My name is Cary Tuckey, I'm a 6th grade math and science teacher at North Kirkwood(?) Middle School, in Kirkwood, Missouri, which his a St. Louis, Missouri. We use the STEM project which is Mathematics.

JW: STEM project is a 6 through 8 mathematics project, it's one of the five National Science Foundation funded middle school programs developing middle grades curriculum to help teachers implement their NCTM Standards.

CT: Who knows the story of Gulliver's Travels? Can anybody tell us about Gulliver's Travels? Allison?

Allison: Well there's this guy and they think he's crazy, but then he goes sailing away and he is on a strange land and he's like really big and these little people, called Lilliputians, they tie him down and they want to make a new shirt for him.

CT: Allison is exactly right. He was all raggedy and they looked at him and though oh, my gosh, this guy needs a new shirt. So they tied Gulliver down and they are doing some measurement. They had to figure out how big a shirt would be for Gulliver. And we're going to see if we can follow their logic and think about how they could make a shirt for somebody that is so much bigger than they are. And I want you to think about something. If a Lilliputian is 6 inches tall, what fraction of your height would that Lilliputian be?

CT: One of the things that we looked at when we were ready to change programs was something that was going to get us away from traditional mathematics books where you teach how to add on one page and you practice it on the next and you never see that again.

JW: The material is presented with a real problems-solving flavor. Students look at big problems and in the process of solving the problem they really develop some of the mathematics concepts for themselves.

Student: The average height of a 6th grader is 5 inches and 6 inches would be half of a foot, so that would be 1/10.

CT: Would anyone like to share your fraction? What fraction of your height was the Lilliputian? Sean?

Sean: 1/10.

CT: Very good. When the Lilliputians tried to measure Gulliver, they used ratios, they used that comparison to make him a shirt. And Allison, would you read for us what they decided would work in order to measure Gulliver's shirt?

Allison: The seamstresses took my measure. As I lay on the ground one standing at my neck and another at my mid leg with a strong chord extended, that each, held by the hand, while the third measured the length of the cord with a rule of a inch long. Then they measured my right thumb and desired no more, for by a mathematical computation the twice around the thumb is once around the wrist and so on, to the neck and the waist. And by the help of my old shirt, which are displayed on the ground before them for a pattern, they fitted me exactly.

CT: If we knew that the Lilliputians were 6 inches tall, and Jonathan Swift who wrote this said Gulliver was about 6 feet tall. What is the ratio of one Gulliver to a Lilliputian. How many Lilliputians is it going to take to make a Gulliver? Pablo?

Pablo: 1 to 12.

CT: That means the Lilliputians were pretty tiny compared to Gulliver. I have a little friend here that I found who is exactly 1/12 of my height. And if we put him down on the ground next to me, can you imagine him having to measure to make a shirt for me? I'm pretty huge compared to him, aren't I? The Lilliputians were smart. They had a formula. And their formula used ratio. They said, that twice round the thumb is once round the wrist and so on to the neck and the waist. What did they mean by and so on to the neck and the waist? David what does that mean?

David: Twice around the wrist equals around the neck and twice around the neck equals twice around the waist.

CT: So we have a ratio, don't we? And if we look at the ratio of thumb to wrist everybody how many thumbs did they say it took to make a wrist?

Class: 2.

CT: Do you believe that?

Class: No.

CT: You don't? What we're going to do is see if it is real. In the box on your desk, you have some string and you have some scissors. I want you to help your partner go once around your wrist and then I want you to actually cut the string so it's exactly the measurement of your partner's wrist. After you do that I want you to wrap it around the base of your partner's thumb, that means right down at the bottom of your thumb. And I want you to see how many times the string goes around your thumb. If you are like a Lilliputian thinks you are, how many times will it be? Twice? Take your ball of string and work with your partner and check that out.

CT: The kids are very skeptical as to whether or not once around your thumb was indeed half of your wrist measurement. When we moved into the first exploration we took string and they used rulers and all kinds of things to verify that ratio and they were very surprised to find out that Mr. Swift did know what he was talking about.

Student: She measured my thumb and then she put it around (?) and then she measured my wrist (and then they put it around my wrist and then we said?) since we have to do it 3 times in a row (phrase?) 8 and then 3 times 2 is 16 and then this is 16, almost 16.

CT: Almost 16.

Student: (?) to 16 (we got around itself?) maybe 16 because you can't take it all the way back to 15.

CT: So would you guys conclude that twice around Lukita's(?) thumb is once around her wrist?

Students: Yes.

CT: All right, very good. That was a good strategy to use. A little different than just wrapping it around.

CT: So how many of you found out that the Lilliputians were correct? You were pretty close. Almost all of you. They were pretty smart, weren't they? Let's suppose then that we have a classmate whose thumb measures 5 centimeters. What would we estimate his wrist would measure. Laurence?

Laurence: 10 centimeters.

CT: And how did you figure 10 centimeters?

Laurence: Because if you go twice around your thumb the Lilliputians say it would equal your wrist so twice, 2 times 5 is 10.

CT: So if we know that the thumb is 5 centimeters and the wrist measures 10 centimeters, how do we say that as a ratio?

Class: 5 to 10.

CT: Would you please take your dry erase boards out and write that ratio two different ways?

JW: We use a spiral approach to instruction without coming back and being refreshed on the material and using the concepts over and over again, students forget them quickly. They need to keep seeing the concepts throughout the year.

CT: I'm glad to see that everybody remembered that not only can we write a ratio as a fraction, we can write a ratio using a colon and we can write a ratio using the word two and all of these are read the same way. How is that?

Class: 5 to10.

CT: Now sometimes we find that it's much easier to compare ratios if we make them into decimals. We know that you can take a fraction and you can easily make into a decimal by doing what? Sara?

Sara: By dividing them.

CT: Good and are we going to divide 5 by 10 or 10 by 5?

Sara: 5 by 10.

CT: So if we take 5 divided by 10, what decimal will we end up with?

Class: .5.

CT: And how do we write that as a decimal?

Class: .5.

CT: .5 good. Let's see if the Lilliputians knew what they were doing when they were talking about ratios? We're going to use our string again and we're going to see if one time around our wrist is half of our neck. So is our neck indeed twice our wrist. We're going to do the same thing going from our neck to our waist. Now, as we measured this time, I'm going to ask you to use centimeters. When we talked about the Lilliputians and Gulliver, we were using inches and feet. But one thing that's nice about ratios is no matter what unit if measure you use, the ratio is going to stay the same. In this exploration I consciously had the kids measure in inches and in centimeters so they understand that if you have a ratio it really doesn't matter what the unit of measurement is. That ratio will stay the same.

CT: If the distance around Gulliver's wrist is 2, what is the distance around his neck?

Class: 4.

CT: And what's that decimal?

Class: (?).

CT: Amazing. And if the distance around Gulliver's neck is 4 what is the distance around Gulliver's waist, would we estimate?

Class: 8.

CT: And that is a decimal of?

Class: .5

CT: Great. What I'd like you to do is take your string and with your partner, measure the distance around your thumb in centimeters, measure the distance around your wrist. I'd like you to write that as a fraction and I'd like you to write that as a decimal rounded to the nearest 100ths.

JW: In the body ratios exploration, students use a lot of different mathematics content. There's a great deal of number sense involved in it looking at ratios and trying to interpret those ratios in terms of a nice fraction and then using that nice fraction to make predictions, they do a lot of work with measurement, collecting data and then interpreting what the measurements mean when they are all done with it. They use averages from statistics to interpret numbers, to make sense out of numbers and to find a number that describes a set. Also use a lot of geometry concepts, just a really integrated approach to mathematics.

CT: Let's look at the ratios that you came up with. And I want you to compare your actual ratios to Gulliver's ratio. What do you notice? Allison?

Allison: They are pretty much the same.

CT: In which form of the ratio did you use to compare?

Allison: The decimal.

CT: The decimal. How many people used the decimal to compare your ratios. And why do you think that was easiest, Cody?

Cody: I think a decimal is easy to compare because the fractions are so different, it's hard to look at each other. So you can compare the decimals easier because they are closer together and you can see if they are together.

CT: Very good. Do you think most sixth graders would have ratios similar to Gulliver?

Cody: No.

CT: No, why not? Lasandra?

Lasandra: Because Gulliver, he is bigger and he's like a giant to us.

CT: But what about the ratio of the distance around the thumb, to the distance around the wrist. Did you find that your ratio was pretty similar to Gulliver?

Lasandra: Yes.

CT: And did the rest of your group find it was pretty similar?

Class: Yes.

CT: And how about the whole class. Did you find everybody's was pretty similar?

Class: Yes.

CT: Yes, they were. So even though Gulliver was huge, that's the cool thing about ratios, that even though he was real big and we're just 5 feet tall, we've got about the same ratio, don't we?

Class: Yes.

CT: Good. Let's just suppose that we now know someone whose thumb measures 9 centimeters. Can you figure out for me how big around the middle that guy is? Talk with your groups and I'd like something on your dry erase board to show me.

Class: 9 times 2 equals 18.

Class: And 18 times 2 equals 36.

Class: 18 times 2 equals 36.

Class: For his neck.

Class: And 36 times 2 equals 72.

Class: 72 times 2 equals 72.

Class: (?) is 72.

Class: And the answer was 72 centimeters.

Class: It's during class the students often work in groups. Sometimes it's a group of 4, sometimes it's just with a partner. There's a lot of discussion that goes on. The discussion is very rich, it's pretty amazing to look at 6th graders and see that they really know what they are talking about. It's interesting to listen to the arguments that they will have to defend their position, to defend their solution, to defend their strategy and they are very good at backing up what they say and realizing that you have to have a reason for what you do.

Student: I noticed on your board you guys have three 2's. Well if you multiply the three 2's together, you get 8. So then if you just take the first one, so it's 9, multiplied by 2.

Student: The exponent is 3.

Student: Yes, with the exponent as 3, it's 72. So it's a little bit faster.

Student: We did the thumb to wrist and then the wrist to the neck and the neck to the waist. So it was just a little bit longer. CT: It looks to me like every single one of you figured out how bit around this guy would be. How many centimeters is his waist, everybody?

Class: 72.

CT: 72, just very quickly, what was his wrist measurement?

Class: 18.

CT: And his neck measurements?

Class: 36.

CT: So that made his waist 72. How did you do that? Sean?

Sean: We multiplied each number by 2 and we kept doing it and eventually we got to 72.

CT: That's a good strategy, but why did you use 2? Why did you multiply by 2?

Sean: Because each time it's half and so if the top one, it has to be half of the bottom.

CT: The numerator has to be half of the denominator?

Sean: Yes.

CT: That's it, very good. Alex, you asked me an interesting question earlier, what was that question? Share it with the class.

Alex: Well, if I was wondering if like, if 2 times around your big toe was once around your ankle.

CT: Let's have some quick predictions. Do you think that the distance around your toe is half the distance around your ankle? How many of you say yes, I think that that's a ratio that is true?

Class: (?)

CT: And how many of you aren't so sure that that would work out?

Class: (?)

CT: What I would like you to do tonight is to go home, you can take string with you if you want to and I would like you to try that toe to ankle ratio with yourself and at least one other person. If you can get data on more than one person, that's fine. And we'll see if Alex has something there. Would you please take all of the things that you've used today and put them in your boxes before you leave class.

CT: One of the things that surprised me was that there was a student who was already wondering weather or not the ratio from her toe to her ankle had any relationship to the ratios that we were already talking about. Another student was using exponents to do some figuring and we've studied exponents, but that certainly wasn't anything that I had thought about using. Other students used a lot of different strategies, which is something that this program really allows them to do. We're not studying a strategy so we apply it, study a different strategy and apply that. They are open to using whatever they want and do what will help them.

CT: Yesterday we found out that we have a lot of ratios that are similar. We looked at our thumb and we looked at our wrist and we looked at our neck and we looked at our waist. And today we're going to look at some other body parts and we're going to see if the ratios are the same for a lot of different people. I want you to put your hands out to the side like this, don't whack anybody next to you. Put your arms straight out. Do you think that your reach from fingertip to fingertip is more or less than a meter?

Class: More.

CT: Put your arms down for a minute and shake them out. Why do you think more? Every single one of you said more. What did you use to estimate, how did you figure that out so quickly? Emily?

Emily: Because like from here to here is about a foot and with your eyes you can kind of tell.

CT: So you used a visual estimation. That's a good strategy. Did anybody use a different strategy to say, you know, that's longer than a meter? Tim?

Tim: When you go out in a reach, I think it's about the same as your height and I know I'm taller than a meter, so maybe that can be -

CT: So you are thinking of a ratio aren't you? We're actually going to explore that today so we'll see if you're correct. Today as we compare our reach to our height, we're also going to look at this part of our arm from our elbow to our wrist and what is that called? Emily?

Emily: The radius.

CT: That's the radius. And we're going to look at the ratio of the measure of your radius to your height and we're also going to look at your tibia. The tibia, Justin, let me use you as a model here. Would you stand and kind of cross your leg like that? Kind of on your toe. That's excellent balancing. You are going to be measuring someone else from the bottom of the knee cap down to your ankle. With the top of the lab sheet that I gave you, you'll notice that you have a person number for each person in your group and then you have four more columns, height in centimeters, tibia, radius and reach in centimeters. What I would like you to do is to partner up again, and very carefully measure each others height, measure each others reach, do your tibia and do your radius. Please round to the nearest half centimeter.

JW: Students come away really seeing how mathematics fits, not just in the adult world, but in their lives too. They get really involved in the activities, the explorations, they really make the content their own, because they are discovering, they are constructing it themselves, they are not being told this is a procedure for doing this and being asked to memorize that procedure. They are creating the procedures themselves and when you create it yourself it really sticks with you. It makes mathematics something that students can use.

CT: All right, it looks like everyone has all of your measurements done, and that you've gotten the information from everyone else in your group. What you need to do now is get up and at the bottom of that chart you'll see it says two other groups and you have students 5 through 12. I would like you to get up and go to 8 other people and get their information. Find out what the distance was for their reach and their height and so forth. It's very important for kids to develop a strong sense of number. In this exploration they are able to apply estimation, they are able to look at fractions, to decimals, they looked at largest and smallest, they predict, they look at ratios. And when kids do that, they are using a very high level of mathematics and that's really a strength of this program.

CT: I know you have a lot of information right now, but look at the very first set of data that you took and that is for just your group. If you look at the bottom of your lab sheet you'll see that you have group member 1, 2, 3, 4, and then you'll notice that we have some ratios. We have the tibia to height ratio, the radius to height ratio, and the reach to height ratio. If we have the tibia to height ratio and we want that in decimal form, if you look up at person number 1 and their tibia measurement, Sam, in your group, person number 1's tibia measured what?

Sam: 30 centimeters.

CT: 30 centimeters. And what was the height of group number 1?

Sam: 150 centimeters?

CT: So we have a ratio of 30 to 150. How do I get a decimal from that? Justin?

Justin: You divide 30 by 150?

CT: That's right. If we have the ratio of 30 to150 everybody, what decimal did you get?

Class: 2:10's.

CT: 2:10. Good. What I'd like you to do now is to complete the top of your chart for all four group members, give me a decimal rounded to the nearest 100s for each of the three columns. When you are finished you might just double check with the other people in your group to make sure that you all agree and to be sure that you've done the division correctly.

Student: Anybody have the same rate as (?).

Student: Yes.

Student: That's weird.

Student: It's really close.

Student: Because it just rounds, you know how you round down around (?) these numbers are so close it was just all around (?) number.

Student: But then the (range in height?) is 94, 94, 99 and 97.

CT: What I'd like to do now is go back to Sam's group and get the decimal ratio from person 2, 3 and 4 so we've got a little bit of data to work with here. Lauren, would you please give me your decimal ratio for person 2?

Lauren: 2:100ths.

CT: And David, person 3?

David: 19:100.

CT: And person 4, Alex?

Alex: 18:100.

CT: 18:100. None of those are the same. And just for simplicity sake, I'm going to make 2:10, 20:100, because we know that those are equivalent. If I wanted to talk about the tibia to height ratio of people, in general, I would be really hard pressed to give every single person's decimal. So what I'd like to do is find an average and we have talked about mean and median and mode. So if I wanted a number that would represent all of these decimals, what would I be looking for? Lauren?

Lauren: And average?

CT: What kind of average would you use?

Lauren: The mean?

CT: The mean? Why the mean, rather than the median or the mode?

Lauren: Because it's a generalization of all the numbers instead of the middle number or the most common.

CT: Good, so we're going to use the mean. And that's the next task that I need for you to do. I want you to take the four data items that you have and find their mean so when you get finished you will have a mean for our tibia. It's a height ratio, the radius to height, and the REIT to height.

Student: It's 20 (?)

Student: Divided by 4 equals -

Student: 195,000.

Student: Oh, I didn't round up, I guess.

Student: You rounded up because the 194,000 and if the 5 were higher you'd have to round up.

Student: And then the 14 is - I mean, the next one is definitely 14, because we are allowed to have 14.

Student: I only had 14, because it's automatically 14.

CT: We have some data to work with here. The group that gave me this data, could you please tell me the mean for your tibia to height?

Student: 2:10.

CT: 2:10. Thank you. And 2:10. Again, I'm just going to put that 0 there since a lot of you are rounding to 100s. That will probably help a little bit. When we look at decimals and we look at fractions we know that they are equivalent that they stand for the same kind of thing, sometimes it's nice to work with what we call nice fractions. On this chart, I have put a few nice fractions. You can see they are very easy, 1/6, 1/5, ¼, 1/3, ½ and then 1/1 or the whole number 1. And I've put the closest decimal equivalent. What I'd like you to do again with your group is to talk about which nice fraction is closest to the means that you got and then we're going to share our information and see if indeed the ratios that we've come up with are common to most sixth graders and then we can think about whether or not they would be common to people in general, just like we did with the Lilliputians and Gulliver. So let's go ahead and find the nice fractions for the mean.

Student: She said (if it's under five round up?) I mean under 5 round keep it the same. (Inaudible). And the next one is 15, so it's closer to 17 and so it should be 1/6.

Student: 16/100 is closest to 17/100 so that would be 1:6.

Student: (?) 200 is closest to 1, so that would just be the whole number.

CT: All right, let's focus back up here for a minute. Our whole objective here was to see if the ratio of these different things were something that we all had in common. And now that you have individual data in your group, let's put some data up on the board and see, if indeed, we did find some similarities. So what I'm going to ask you to do is by group, give me first your tibia to height ratio. If you would give me your decimal first and then give me your nice fraction, we'll get that recorded. So group 1, what was your tibia to height decimal.

Group 1: 24:100.

CT: And our fractions?

Group 1: ¼.

CT: So we looked at each of the groups, put them together, looked at the decimal, also looked at the nice fraction and found out that there was a consistent ratio. So with a 6th grader we really could make some predictions as to what these different length measurements would be.

CT: If you look at the data that we gathered on the board, what do you notice about the fractions in each column? Alec?

Alec: They are all pretty much the same.

CT: They are all pretty much the same. And if you look at the decimals what do you notice about those?

Alec: They are all the same.

CT: They are all pretty much the same. I want you to look at the data that we have up here and actually I'm going to say that for our tibia to height ratio that if you look at that we've pretty much found a 1:4 ratio, our radius to height was 1:6, and our reach to height was 1:1. I want you to see if you can predict the tibia length of a person who is 150 centimeters tall?

Student: Okay, I think it would be 37 ½ centimeters, because down here we have one with 38 and 38 times 4 equals 152, so I just used 37 and 5/10 times 4 and that equals 150.

JW: We talk about them constructing their own understanding, but this is in a structured fashion so that the teacher's monitoring what is happening all of the time to make sure that they are in fact, learning addition and fractions they are learning how to divide decimals. But they are also learning at the same time to use and to apply this material.

CT: If we have a person with a height of 150 centimeters, and we want that tibia prediction we know that we have a 1:4 ratio. What did you do to predict the length of that tibia. Emily what did your group do?

Emily: Well we figured that if the person is 150 centimeters long that it's a 1:4 ratio that you should divide 150 by 4 and we got 37.5 and we weren't sure if we should round it or not so we left it there.

CT: When it comes out to exactly 37.5 then there isn't any reason to round so you guys did exactly the right thing. Did every other group get 37.5?

Class: Yes.

CT: 37.5. So we have a man or a woman 150 centimeters tall with a tibia length of 37.5 which gives us that 1:4 ratio. And now comes the test. Do you remember those Lilliputians that we started out with yesterday? How tall were they?

Class: 6 inches.

CT: They were 6 inches tall. What I would like you to do is with your partner, draw for me a 6 inch Lilliputian, a stick Lilliputian is fine, we don't need lots of fancy things, just a sick Lilliputian. And using our class ratios because we were so close, we had everything almost identical. I'd like you to label for me the length of the Lilliputian's tibia, the Lilliputian's radius, his height, and his reach.

CT: At the end when the kids had to draw a Lilliputian and actually give a prediction as to what the measure of the tibia would be, the radius, etc. they didn't have any problem. They really understood what the radius was, they could very easily transfer that use, use the algorithm, and apply it to something that was a little bit different.

Student: We found out that the length of the radius was 1 inch.

CT: And how did you get that?

Student: Since the radius is a 1:6 with the height we divided 6 by 6 and we got 1.

Student: For the tibia, there was a 1:4 ratio so we did 6 divided by 4 and we got 1 ½.

CT: 1 ½ inches, all right. And how about the reach? How did you get 6 inches for the reach?

Student: Since the ratio was 1:1 and since the height was 6 we just made the reach 6.

CT: All right, you have a great-looking Lilliputian there.

Student: (?)

CT: We know that your Lilliputian was 6 inches tall. What was your prediction for the length of his tibia? Allison?

Allison: Okay, since the ratio is 1:4, I just took 6 and divided by 4 and I got 1 ½.

CT: What was your prediction for the radius? Bailey?

Bailey: I predicted that it was 1 inch, because of the 1 to 6 ratio.

CT: What did we do for the reach, Cody?

Cody: I did 6 inches because your reach should be the same length as your height so I just made our little man 6 inches.

CT: That's right, the reach of the Lilliputian was 6 inches. Now that Lilliputian is kind of a tiny guy and I want you to imagine Gulliver who is 6 feet tall. So if we had the Lilliputian and now we have Gulliver, Gulliver being 6 feet tall. If a Lilliputian's tibia is 1 ½ inches, what is Gulliver's tibia? What does that measure? Emily.

Emily: 1 ½ feet, because 6 divided by 4 would be 1 ½.

CT: We still have the same ratio, don't we?

Emily: Yes.

CT: So let's put 1 ½ feet. Now that we know that everybody, you have the Lilliputian has a 1 inch radius, what would Gulliver's radius be?

Class: (?)

CT: And if he is 6 feet all how about his reach?

Class: 6 feet.

CT: How many of you have a dad who's 6 feet tall? You can go home and impress him tonight. Tell him, I can just predict your tibia and your radius and you can amaze him with your mathematical skills. But I want you to amaze me one more time. The average 6th grader is about 5 feet tall, come here, Clayton, I think you're about 5 fee tall. If Clayton is now our Lilliputian how tall do you suppose Gulliver is? Emily?

Emily: 60 feet.

CT: 60 feet. That sounds huge. And I think tomorrow we need to go outside and see how big 60 feet really is. Yesterday we finished up an exploration and we were looking at the height to reach ratio. We talked about us actually been Lilliputian size. And an average sixth grader is about 5 feet tall. When you are a Lilliputian and maybe we should label this so we know that these are our Lilliputians and you are 5 feet tall, if it takes 12 of you to make a Gulliver, how tall is Gulliver now?

Class: 60 feet.

CT: And if he's 60 feet tall, what is his arm span?

Class: 60 feet.

CT: And if I'm going to make a shirt of Gulliver's and I want to go from finger tip to finger tip so it doesn't get cold, how long do those shirt sleeves have to be?

Class: 60(?) feet.

CT: They sure do. Now would you please sketch a real quick sketch of Gulliver's shirt and since we already know his arm span let's mark that at 60 feet. If you look at the picture on page 401 in your book, you will see a girl who is standing with her shirt just like your picture of Gulliver's shirt. If you looked at just one arm, what fraction of her whole reach would that one arm be? Melissa?

Melissa: It would be 1/3.

CT: It would be 1/3. If we have a 60 foot arm span what is 1/3 of that arm span going to measure?

Class: 20 feet.

CT: So let's go ahead and say now we know that each sleeve is going to measure 20 feet, we know the body of that shirt is going to be 20 feet and we can get all of those measured. What would the measurement for the body of Gulliver's shirt be?

Class: 20 feet.

CT: 20 feet good. So let's go ahead and get that measurement in there. And now we have the two toughest measurements. Justin, you have a long-sleeve shirt on, so let's use you as our sample here. This is our miniature Gulliver. We're trying to figure out what the distance is going to be from Gulliver's arm pit to the base of his shirt to the tail of his shirt. So if you look at, Justin, I'm going to make you Gulliver here. There's an arm pit and there's a sleeve. How long did we say the sleeve is?

Class: 20 feet.

CT: Put your arm down to your side. What can you tell me about the length of his shirt and the length of his sleeve, Sam?

Sam: It's about the same.

CT: So we have another 20 feet from Gulliver's arm pit down to the tail of his shirt and we'll just measure one side, and we'll assume that we're great seamstresses and we're going to make it all even. What's the one measurement that we have left to do? Sara?

Sara: The arm cuff?

CT: The arm cuff, that's right. It's a little difficult to look at my lovely drawing and figure out exactly the size of that cup. But look at that girl on page 401 again. Let's look then at her sleeve and see what fraction of her shoulder to the tail of her shirt you think that sleeve is? Clayton?

Clayton: I think it's ¼, because I use my fingers, and like the width of this, you can like take it down 3 more times.

CT: So you're saying that that's ¼ of our total height. Great. If this is 20 feet, that means that this is ¾ of the total. If this is ¾ and this is ¼ does anyone have an idea of how we can figure out what that ¼ measurement is? Melissa, what do you think we could do?

Melissa: You could divide 20 by 3 to get the answer.

CT: Okay, and we divided by 3 because?

Melissa: It's ¾.

CT: Good. So what would the measurement of the cuff be?

Melissa: About 7.

CT: About 7 feet, very good. Let's go outside and see what we can do.

CT: One of the things that some teachers were apprehensive about when we began this program was, it's so different, I've got to have all of this stuff, I'm going to be running around, what is it like? There's a lot of teacher support material. You have lists of things that you need and most of it you have in your classroom anyway. There's a wonderful resource book that gives you very good instructions and ideas as far as how to carry out a lesson, so the teacher support materials make the jump a lot easier. On your groups clip board, you have some instructions for measuring and what we're going to do is start with our mid point stake which is right here by Lasandra, that's a mid point of the neck of the Gulliver's shirt. So if Group 1 would come and start with Gulliver's sleeve, if the rest of you would just come back here, we're going to get them started, remember we're going to get them started, remember you want to do some sighting and help them out to make sure that everything is straight. We've got some stakes that we're going to put in the ground and when we're finished with that we'll actually put some crepe paper down and see if we can outline Gulliver's shirt, we'll go up on the hill and see how big it is.

Student: We'll have to go from the mid point to the bottom of this shirt which will be 27 feet and we'll place 1 stake there.

CT: Very good. Does somebody have a stake? You guys, who remembers what perpendicular means. What's perpendicular mean, Makita?

Makita: A 90 degree angle.

CT: A 90 degree angle. Who is at a good vantage point to tell you whether the sleeve reached and that center line is perpendicular? (?) Probably Sara is. What does Cody need to do?

Sara: Cody, you need to go to my right.

Cody: A lot.

Sara: Yeah.

CT: One of the things that's really nice about this being a program where the kids are so actively engaged is that it's very easy to extend and come up with ideas that the kids really enjoy.

JW: The real satisfaction from developing programs of this type is when you actually see students using it, but the enthusiasm that comes through, they are loving what they are doing and they are taking our ideas and they are extending them beyond what we had ever imagined. And that's the real thrill of working with a program of this type. And I think it's a real thrill for teachers too.

Students: We need 1, 2, 3, 4, 5, 6, 7, 8, 9,10, 11, 11. We are Lilliputians.

CT: We are Lilliputians. (?) you want to go up on the hill and (?) okay, go ahead.

CT: It's very rewarding to see kids do something that you love so much and enjoy it and realize that they can do it, they were all actively engaged, they were engaged together, and they were on task, they were talking math. They weren't doing something because I asked them to do it. They did things on their own. As a teacher, anytime that you can get that feedback from kids you go home saying, that was a good day, I think I'll come back tomorrow.

JW: 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. END