TERRI DEJA: I'm Terri Deja and I'm 6th grade teacher here at Buchanan Middle School in Buchanan, Michigan. I've been teaching math and science here for six years. This is my 20th year of teaching. My teaching's evolved a lot since I started with Pathways. I started out just strictly using a traditional textbook. And once I was introduced to Pathways I saw a lot of engagement with the kids, they would grasp concepts, they'd hang onto them, and I thought, this is just the way I gotta go.

TD: Do you remember about three weeks ago or so when we got together and we were doing our blueprint dream home? We had decided on who your client was and where exactly you were going to be building this home, and then we took that time and you put together a blue print on the computer.

TD: The name of the unit is Architect Dream Home. Pathways is structured so that the students take on a role in their role as an architect they have to create a client that they want to build a dream home for. The way I start out the program is I take the kids around with the hand-held video and we just do a quick walking tour of the area, all the kids think it's really fun, they get to see themselves on camera and then they get to use that to incorporate those interesting details that we noted on the tour into their dream home design.

TD: First I have the students design their dream home on grid paper. From there what we do is we go to the computers and we begin to sketch their dream home blue print.

JENNIFER KNUDSEN: Pathways consists of a series of design units, most of those last four or five weeks in the classroom and each one of those is centered around a design problem. In the unit dream home, the students use architect which is a computer-aided design system. The software allows the student to build one, two or three floors. They can build walls, doors, windows, they can drag any of the furniture to a different location. They can also create a data window to calculate the building costs.

TD: There's something I want to talk about today and that's about floor space. This is one of those floor plans that we had looked at that we had gotten off the Internet, and in this design we noticed the bedrooms and the storage area. What I want you to be concentrating now on today is the study area. I want you to take a look at the print that I'm going to give you, I'll give you each a copy of this in just a minute and what I want you to do is figure out what is floor space?

TD: I give them a blueprint that is identical so that everyone is using the same rectangular shape to find floor space.

And materials are up in the front, you can help yourselves to whatever is going to make your job easier finding floor space.

JK: Our program is designed to meet all of the NCTM principles and standards. We really began in our development by turning the design process on its head. We started by looking at adult professionals doing their work. We saw what kinds of mathematics they needed to do and then we took a slice of that real-world work and cut it down so that it fit into the middle school classroom and then highlighted the mathematics that was part of the middle school curriculum. In the evaluation that was done of our draft materials we got extremely high scores on student engagement.

Student: We don't have a scale on here, so you need to get a scale for where the carpet is.

Student: How about 1 centimeter equals, 1 square equals 1 square meter?

Student: So we're going to have to measure right?

Student: We're going to have to measure centimeters so I'm going to measure it. 3 ½, 5 ½.

Student: So make a sketch of the room so you know (?) figure out a scale?

Student: How about we make every square centimeter equals a square meter on the scale, for the scale of the study room.

Student: What do we do with the little one?

Student: Well the little ones, well so many little ones will make up a whole one, so we can get two more centimeters out of those if we add them up.

Student: How do you know that?

Student: Well we have 3 ½ by 5 ½ so we'll have 15 whole square centimeter units if we halve the whole ones, but the halves, every time you add two of the halves together you'll get one more and there are quite a few halves that we need to add up, so we can get a few more out of those.

TD: Okay, group, let me get you all back together. You've been spending a lot of time on this and I've been seeing a lot of really neat ideas. I'm going to ask Jessica and Paul to come on up in the front and show me what your strategy was. There were a lot of different things going on, a lot of good math talk, it should be interesting.

Student: Okay, we used one block equals half a meter, so we took the blocks and we put them in and we came up with -

Student: Yeah, but we used little blocks on our other thing.

Student: It was like really cool, because it was really cool.

TD: Okay, so when you had this flat on your surface you were putting in blocks. But I was talking about floor space. Where did you come up with the idea of blocks?

Student: Well it would cover the whole floor and we thought that's where like (?) was.

TD: Could you use circles?

Student: No, because there's no edges to the circle so he couldn't put them in the corners or -

TD: Okay, so if put a penny up here and laid pennies across here, you are telling me that wouldn't work.

Student: There would be open spaces.

TD: Open spaces in between. All right, now I understand why you did the blocks. Questions, Sean?

Sean: When you guys used the blocks did you consider counting the wall spaces as in how thickness of the walls, because if you counted the thickness of the walls it might be a different area, of floor space.

TD: Different floor space. Did you?

Sean: We didn't (?) that.

TD: So show me exactly with your finger what you considered floor space, outline it for me, please.

Student: But this might be a bookshelf, you said, so -

Student: But it would have to sit on the floor because it's not hanging in the air.

TD: Okay, if that was a bookcase then it was sitting on the floor and you'd have to have floor space underneath that. Okay.

Student: One of the strategies we used is we did length times width so we measured -

TD: Length times width. We were using floor space.

Student: Well it's to find the area of how much room there is in your, inside of your room.

TD: What unit of measure were you using?

Student: Centimeters. But that was on ours and ours was littler than your blueprint. So that would be different.

TD: You'd have to take that into consideration. Okay.

Student: And then we took these blocks and we put them across and ours is a little (?) so we had four across here and six going down, so in the middle we had 4 and 8 would be 24.

TD: So it was kind of a check.

Student: Yes.

TD: Okay, and then you came up with 24 what?

Student: 24 in the middle, so that would be our flooring.

Student: Okay, you'd need 24 little squares in the middle to be the flooring.

TD: Yes. Okay, interesting. Now since we've talked about all these different strategies, I kept hearing things like area, that word kept creeping in. So I'm guessing maybe that's something we need to talk about. What is area? Ashley?

Ashley: Length times width.

TD: Length times width. That sounds like a formula, like when you mix chemicals together or something like that. What do you mean, length times width? What will that do for you?

Ashley: Length times width finds out how much area is inside the room.

TD: So that's a real key word, inside, when we're talking about area. We talked about length times width, that's the formula Ashley, I think you use, well what do you get when you multiply length times width? Ryan?

Ryan: When you do length times width, you get the total space inside the room.

TD: Okay, but if I said the length was 3 times 2, the answer then would be -

Ryan: 6, whatever your scale is, like squared meters, or squared -

TD: Hold that thought. You said some really key words there. You used the word number and square units and scale. Those are all important things when you're talking about a definition. So we've got insides a figure, and we've got square, we've got units that are centimeters, meters and inches. Are they centimeters, inches and meters? Stanley?

Stanley: No, they are actually squares. See like, see how the big the room is and see how big your squares are and -

TD: And so are you telling me I need to add something to this, cm?

Stanley: Squares, centimeters would be the answer.

TD: Okay, so I could put the word square in front of it. Is there another way that I can identify centimeters being squared? Joe?

Joe: Yeah, you can put like a little 2 next to it, or after it in the air.

TD: That works, remember doing that? Centimeters squared, meters squared, inches squared? Okay, using what we've got on the board, can you come up with a definition for area. Not a formula, we've already talked about that, what about a definition for area? Amanda?

Amanda: Area is the amount of space in the room.

TD: In the room. Could someone then use another couple of these words to add to Amanda's beginning? Sean?

Sean: It's how many square units will fit inside of a room that you would use?

TD: That's great. How many square units -

I think it's very important that I use the kids words and their terminology to come up with a definition, a workable definition, a definition that they understand and then, from that I'd keep pulling until I'd get exactly it worked and worded so that it's a mathematical definition. Jeff?

Jeff: That will fit into the figure.

TD: Okay, so if we explained this to a third or fourth grader, by using the definition, the number of square units that will fit into the figure, do you think they'd get a clear understanding of what are is, rather than using length times width? Ryan?

Ryan: I think it will be a lot easier for them to understand than just throwing length times width in their face and then they probably don't even know multiplication yet, so if you told this to him this would be a lot easier for them to understand because they'd have the blocks in front of them and they'd be able to piece it on the object or the figure and it would be a lot easier for them to understand.

TD: Good point. I think it's a lot easier for you to understand, it's a lot easier for me to understand and explain it to you -

After we have worked through our definition then we're actually working with the kids own individual designs.

TD: We just finished talking about how you find the area of a regular rectangle. But what happens if you have an irregular room?

This lesson focuses on measurement, specifically on the concept of area and then how to measure the area of an irregular figure and finally an open-ended problem that includes area in the solving of it.

TD: Okay, what you have in front of you now are the drafts, this is probably your third or fourth draft and we've got a long ways until we get to completion. What I want you to do though now is look at see if you can find an irregularly shaped room, something that's not a rectangle and I want you to find the area, we've got a definition for area, the number of square units that will fit into the figure. You have rulers, you have cubes, you have lots of different things, how are you going to find out what the area of that irregular room is. The other thing I want you to think about is, taking notes as you're doing this, because I want you to be able to explain it to other groups. Okay?

Student: How(?) we do it?

Student: We're trying to make a rectangle so we can take the sides and subtract if from 28 which we got from adding the length and the width.

Student: Length times width.

Student: So we can just subtract the sides that aren't part of the room and we can find the area of those sides.

Student: Why don't we number them?

Student: We can number the halves like to make one whole. So this is 27. And then here's two more squares. So it's 28.

Student: But there's 28 in here.

Student: There is 28, so this is 29.

Student: 28 with the (?)

Student: Subtracting it

Student: Now what about this side?

Student: It's not right.

Student: It's not on the dot.

Student: Especially that one.

Student: We're going to have to break it up maybe, but how?

Student: This plus this would be 24. This and this is a whole, it's one whole square. So that's 24.

Student: What about 28? Oh, wait, not what about this?

Student: Wait, put that, this and this and then this corner up here, so it would be 23, 22. So we have areas 22.

Student: The square what, is it going to be meters or -

Student: Meters probably.

TD: So have you figured out the area of that unique room?

Student: Yes.

Student: 22 square meters.

TD: 22 square meters. Tell me, wow, how did you do this?

Student: Because if we put that corner right there it would be about (?) square.

Student: But first we had to put the rectangle around the room and subtract those sides.

Student: And then we just had to number it -

TD: That was a good idea to number it, a real good idea, because we could keep track of it.

To find the area they started to subtract and I had mindset that they were going to be just subtracting and low and behold they switched gears and they were actually counting backwards using a totally different algorithm than what I had in mind. Things like this crop up all the time. I'm always surprised when kids are finding different ways of doing things and that's what makes this program so unique.

Okay, I saw a lot of neat stuff going on in here. A lot of different strategies, you were all using that area is the number of square units that will fit into the figure information to find out your area for your nonrectangular room. I'd like a group to share with us what they've got and Ashley, Jacquelyn, Heather, the three of you had a really large area, come on up to the board and share with the rest of the group how you did it. Before you get going, outline with your finger the room that you're working with. Wow, wait. How did you get going on that big area?

Student: Okay, first we did it the hard way. We counted all these little boxes through here and we got 102 for the area.

TD: Then what did you decide? What made it easier?

Student: We counted the, we divided the, the boxes on the edges -

Student: We divided them into rectangles, so they were easier, so they could do length times width. So we could find all these in here.

TD: So start with the top section.

Student: We could write that into a rectangle and then, well we did this the hard way too. We added them up, but it's 3 times 6, which is 18.

TD: Did the rest of you see how they did that/

Student: Yes.

TD: Did any of you use that strategy on any of your rooms?

Students: No.

Student: All right, let's continue then, because you've got some small, little pieces that stick out there in the corners. Jacqueline, what did you do with those?

Jacqueline: We made them into the boxes, like we just divided them off right here, and then we did 1 times 3, which is 3, on each one of the and it equals 3 and 3 square meters.

TD: 3 square meters, good, I'm glad you're talking in those math terms. Heather, then that bottom section?

Heather: We multiply 3 times 6, which is 18.

Student: This is the main room of it and it is 12 times 5 which is 60 and 60 square meters and what we did is we multiplied them together to get 60.

TD: So not that you've got all those rectangles and you found the area of each one, what's next?

Student: We added 60 plus 18,18, 3 and 3 and we got 102/

TD: 102 what?

Student: Square meters.

TD: Square meters. Does that sound reasonable?

Student: Yes.

TD: Okay, I really like this diagram. I like the idea of enlarging it like this, but I noticed when we enlarged it that things got really big. Do you notice anything about proportions or -

Student: The toilet is -

Student: Huge.

TD: Like show me with your hand how huge it is it.

Student: (?)

TD: Wow! Big toilet! Anything else?

Student: And also I noticed that the bathtub was the size of our bed.

TD: How many of you have a bath that was the size of your bed? Any of you? Okay. Everyone remember that this is just a draft. All of you are still in the draft mode, nothing is finalized, because we've got a couple of more weeks to work on this. Okay, thanks, you did a nice job ladies.

TD: I brought Mr. Scott from the local carpet company and he's a carpet installer, he's going to be talking to you about how to install carpet and what to look for. We are going to be doing that in our dream homes and he's got some pretty good tips.

Mr. Scott: Good morning, guys, who has carpeting in their house? Everybody.

JK: The program is a complete package that teachers can pick up and use, just as it's written, but because of our focus on keeping mathematics embedded in the real world, there are really a lot of opportunities for incorporating community involvement into this program.

Mr. Scott: Can you buy carpeting as wide as this room is?

Students: No.

TD: I really like to bring in outside resources.

Students: Where's a good place to put carpet seems?

Mr. Scott: That's a good question.

TD: It gives the kids a chance to hear from someone else besides a teacher.

Mr. Scott: The seams are going in this direction and you need a lot more carpeting if you were to lay it this way.

TD: They have this concrete example of how carpet is installed, we're modeling the students laying carpet in their dream home.

Mr. Scott: You'll notice that there are some big roles over there, that kind of look like carpeting. Actually they are wrestling mats, but what happens is carpet comes in big roles like that, but when you put it down on the floor you've got to make it look like one single piece that was designed just for that room. That's why it's important to where you put the seems, and you know how much area you've got, because that tells you how much carpeting you need to buy. So one thing you need to consider when you're buying carpeting is the length of carpeting that you need, because you buy it in a big role and you are going to cut it in sections and then seam it together. The other thing you want to consider is where the seams are, so it looks good and they are not going to be on a spot where people are walking all over them and making them wear out.

TD: Thanks Mr. Scott. It was nice of you to come today, and ladies and gentlemen, thank Mr. Scott for giving us some time, great tip. Wow, you just learned a whole lot from Mr. Scott. We talked about how you have to buy carpeting in rolls and you have to determine the length. I have a job for you. You are now carpet installers. And you are installing carpet in your irregular room that you just measured the area on. Here's what I want you to do. Because you've all got great ideas, I want you to come up with two different strategies to use to find out how you would lay carpet in that irregular room. I'm going to give you a roll of carpeting, you have unlimited length, but remember, cost is a factor for you and your clients. So you need to keep that in mind when you're determining the length. Two strategies again, I want you to write them down, because you're going to pick out which one at the end is the best and explain to me why you chose that one. Okay? Carpet!

The students will have a strip that represents a width of 3 meters.

Okay, if you need more you can purchase more -

JK: By bringing real-world settings into the classroom, we're telling kids, use everything you know about carpet. Use everything the local guy at the carpet shop told you about carpet, and apply that to the problem that you're working on.

Student: You've got 13 of this - 13 meters long.

TD: A length?

Student: In the length. And we have one seem right down the middle.

Student: It's covered up by that.

Student: Table.

Student: He said it wasn't good to have it right down the middle, but the dining table's going to cover the seem up.

TD: So there won't be a lot of traffic.

Student: Yeah, and people aren't going to notice it very much, but we also found another way to do it and it's 18 meters long so you're going to have to buy more carpet, and you're even going to have more seems too, so we think that it would be better to do it the other way. And the other way it was 13 so if there's - when you match this up, you have to buy a lot more carpet and if you did it, this is the better way, because if you did it this way then you're not going to have to buy as much and you're not going to have as many seems.

TD: Less carpet, less seems.

Student: Less carpet, less -

Student: Less seems, less money, cutting and you kill a bunch of birds with one stone.

Student: And if the clients prefer the other way this fits like perfectly under here, except it didn't, because we didn't cut it too well, but -

TD: Can you think of any reason why the clients would like it the other way?

Student: No, but I mean, --

Student: (?) they (?) if they did.

TD: Okay, so to give the clients a choice.

Student: I don't know why they would, because they probably want to spend less money, but -

TD: And because they (?)

Student: Yes, this is definitely the better way.

TD: Good, good efficient work here, nice job guys.

JK: There's a real sense of ownership that comes with design and from that sense of ownership comes a kind of enthusiasm and perseverance, that can be hard to engender in the math classroom.

Student: (?) 3 ½.

Student: They are right on, 2 ¼.

Student: Look, this is a quarter right here.

Student: It's not right on it.

Student: So we're going to have to draw this.

Student: So we're going to have to draw this.

TD: Sometimes when the kids were working I can sense that there's some frustration going on. I watched the intensity building.

Student: That needs to be 1 ½.

Students: Let's just make a new piece.

Student: No, Derrick, why?

Derrick: Because this piece here is too small.

TD: If I need to intervene, I will, but quite often they are getting real good at solving their own dilemmas.

Student: Cut it into 3 pieces?

Student: (?) carpeting (?).

TD: Okay, now what I have for you is the transparency, and you may need one or two, or maybe in three, depending on the size of your regular room that you're going to be copying. You've spent a good time trying to figure out what two strategies you are going to use to find the area and order carpet. On this transparency, I want you to trace your irregular room, I also want you to list how many meters of carpet you need to purchase. You have to write down the length of carpeting, also make sure on this transparency, you draw in your seems. Then one of you will be responsible for writing up your two strategies in your journal and I'll be collecting that as well.

TD: When the students are in groups it's hard to assess whether or not each one is grasping the concept and I can't be everywhere at the same time. So what I can do is take a group example such as the transparencies and their explanation and then I can make sure that they are on target. I spent enough time listening, but it gives me something in my hand, it helps me to see where the kids are headed, if they've grasped the concept.

That will wrap it up for today. You guys did a lot of nice work, I'm proud of you.

Today, ladies and gentlemen, since we've got the computer lab, what I want you to do is pull up your blueprints of your dream home and, your main floor, and we're going to be printing a copy of this because what we're going to do today is work on scale. I want you to remember that in order to change scale it's this little box that you unlock and this is where you can change the scale to a half. Last time when we started with architects you had all of your scales set for one meter in the real world. And if you were to draw the bench, then -

Scale is a real important factor when we're dealing with the whole architect program. And important concept that comes from actually looking at their blueprint and changing the scale is that we have a drawing that's exactly the same size, and exactly the same shape, but because the scale's different it represents different length.

Okay, you're going to be choosing one of your rooms to enlarge, because what we're going to do eventually is rearrange the furniture in one of those rooms. And it's very small on the screen, so the first thing I want you to do is print out one of your blue prints of your dream home design and you're going to use that then to model your second drawing which is going to be an enlarged picture of only one room.

JK: We think it's absolutely essential that kids get significant exposure computer use in the core curriculum and so we designed our program to make sure that all kids who are learning mathematics are going to get access to learning mathematics with computers. It's not that you need to a lab full of computers to do this program, but the computer is an important tool to be used for design and analysis. This technology is designed not only to be kid friendly, but teacher friendly too. Of the four pieces of software that we've developed, each one is reused over the two year curriculum, so that kids get a chance to get all of those features, and they don't have to relearn them.

TD: You and your partners now decide on one room that you're going to enlarge and arrange furniture. What I want you to do then, is to change your scale on the computer and then from there, do another drawing that's enlarged of that one single room. After that's been done and you've checked with me, then you can start drawing in your furniture. Please remember, as you're putting in the furniture that your scale has changed.

Student: We're going to change this to .25.

Student: And we're going to lock it up again -

Student: What room are we going to (?).

Student: Let's take -

Student: Let's do this one.

Student: Okay, we'll do that one.

Student: Okay, we're going to go 1, 2, 3, 4, 5 - 4, 20.

Student: (?) 4, but that's 5 meters.

Student: (?) 20 squares which is 5 meters still.

Student: How many?

Student: 20.

Student: 20.4 all together.

Student: That's only 4.

Student: 1,2, 3, 4, 5, 6, 7, 8, 16, 17, 18, 19, 20.

Student: Okay, now how many down? How many is it down there?

Student: 4.

Student: No we just go 1, 2, 3, 4 -

Student: Just five dots.

Student: 5?

Student: 5.

Student: Yes, because 1, 2, 3, 4 - 24.

Student: 24, so it's only 16 dots now.

Student: 13, 14, 15, 16.

Student: And it goes across.

Student: It's going to be the same.

Student: Just going straight across.

Student: That's a big room.

Student: (?) ask you about the furniture or (?)

Student: We are going to add the (?)

Student: How big is the furniture?

Student: How big is the -

Student: The bed is 1(?) meter by 2.

Student: Holy cow! Wow that's a big bed.

Student: No it's only a 21 x 2 still.

Student: These are only ¼.

Student: (?) 1 x 2 beds still, it's a small bed.

Student: For you it may be big, but it's actually in real life very small.

Student: That's 1 meter, right?

Student: Okay, next is (?).

Student: That's a TV stand right?

Student: It's 4 x 4.

Student: 4 x 4.

Student: 4 x 23(?).

Student: That's a normal size room.

TD: What's wrong with it Heather?

Heather: That's too big.

TD: Too big, too long and it's how many meters?

Students: 2 x 1.

TD: Well, let's pretend the floor is our bed. And let's see if you'll fit in a 2 x 1 meter bed. Now, okay, Heather pretend the wall is your pillow. What do you think? Is that about a meter? Put your arms out and see if that's about 2 meters in length. Would that work?

Student: (?).

TD: Okay, Stan what about the width? Is that about a meter wide?

Stan: Yes.

TD: Where is? Okay. So even though that the picture looks bigger on your enlarged room design, in actuality I think it's just perfect for a single bed. Heather fits in a single bed. 2 x 1 sounds good to me. Katie, you have a question?

Katie: I don't understand this, because this measurement that we just did looks exactly like our floor plan.

TD: Oh it does, the floor plan looks exactly like what you have on your screen. Tell me what you change your scale to?

Katie: 2.5 meters.

TD: 2.5 meters. Okay, so you increased your scale.

Katie: Yes and we thought if we made our scale bigger that our drawing would be bigger.

TD: Your drawing would be bigger, but it didn't work that way?

Katie: No.

TD: Now, each one of those little squares measures how much?

Katie: 2.5.

TD: And what did it measure before?

TD: 1 meter. Okay, so a little tiny square used to be 1 meter, and now it's 2 ½ --

TD: 2 ½ okay so let's think about 1 meter. So this is how much 1 meter was and now you've increased it to be 2 ½, which is like from here to the wall. So what's happening to your drawing?

Student: It's getting smaller, but we think it should be getting better.

TD: But you're supposed to be enlarging it, right?

Student: Yes.

TD: So what's going to happen to the scale then if you want your drawing to be larger your scale will have to -

Student: Be smaller.

TD: Try it. Let's see.

Student: Like .5.

TD: Okay, which is ½. Now, every little section that you had there that used to be a meter is now -

Student: ½ meter. So how many of those little sections do you need to make a meter? Oh, so you had to make more sections, so you like have to double it - we used that to have 8 dots, now we have to have 16 dots going one way.

TD: Exactly, you got it. Is that going to work? Try it.

TD: It is very difficult for kids to understand scale sometimes. The students think that as they make the scale larger that their drawing is actually going to be larger, that's when you just take that approach and let me discover and talk and talk and talk until they realize that the smaller the scale, the larger their drawing will be. That's the beauty of learning math in this way. Students are actually getting the experience and the problems are generating from them, they are not just teacher-driven problems, they are coming up with them on their own and finding ways to solve them.

TD: Okay, so what I want from you on Friday is your enlarged drawing and your regular design. Okay? See you all.

TD: I just love this dream home program, because what it does is it opens up so many avenues for the students, they get a chance to do a lot of problem solving while they are learning loads and loads of math skills and it just makes me feel great when I'm watching kids go, oh, I know that or gee, this is so cool, or wow, why did that happen that way. I even tell them, I said, you know when you're in high school you'll think back and you'll remember, architect, man, that's where I learned all about area.

For further information about the Modeling Middle School Mathematics project, please contact, www.MMMproject.org.

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

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