My name is Lisa Brown, I teach 7th grade math and 8th grade math at Keiling Junior High School in Austin, Texas.

LISA BROWN: Okay, good morning. Today our goal is to focus on finding squares. We found some squares a few days ago, but today I want to focus your attention on some squares that only will fit on what I'm going to call a 5 x 5 dot grid, I'll show you what I'm talking about.

What we're going to be doing today in this lesson is having students finding squares and developing an early understanding of the concept of square root as the length of the side of a square.

So the first square that I could draw would just be one small square right in the corner. Can someone tell me the dimensions of this square? Ashley?

Ashley: 1 by 1?

LB: And Ashley, do you know the area of the square?

Ashley: 1.

LB: So it's a 1 by 1 square and its area is equal to 1. Okay, so that would be 1 square unit. Can someone tell me another square that would fit on a dot by dot grid, another square that would fit? Dominic?

Dominic? 2 x 2.

LB: Great, thank you. And do you know the area of that square?

Dominic: 4.

LB: Dominic, can you tell me why you knew that the area of this square was 4?

Dominic: Because I took 2 x 2 and I got 4.

LB: How else can I know that the area of this square is 4? Christina?

Christina: Because you could cut it into 4 squares?

LB: 4 square units. I want you to find as many more squares as you can that would fit on a 5 x 5 dot grid. Label the area of each square that you find. Using some of the strategies that we've used in the past few days for finding areas of squares. I want you to work with a person right next to you. We're going to be looking for squares, labeling their areas, I'm listening for conversations about how you guys are finding these squares and how you are finding the areas.

Our particular feature of this lesson is that we're tying in number with geometry and tying both of those together into a physical model and that's a feature throughout the connected mathematics project that we're making connections across different mathematical subject areas.

Student: 3, 6, 9, the area would be 9.

Student: All the way down -

LB: Francisco, you had a question, what was your question?

Francisco: I wonder if you can a slanted square?

LB: A slanted square? Yes, you can make a slanted square. You guys remember the investigation we did where we were trying to find those parks for the city of Euclid?

In an earlier lesson I had given the students a map of fictional city called The Euclid? And the premise of this lesson was that the City of Euclid is wanting to build some park land and I'm asking students to find parks in different shapes and one of them is a square, a tilted square that leads perfectly into this lesson. So think back to the strategies that you guys were using then and see if you can some of those same strategies, so that like I said, some slanted ones would fit on here.

It's very interesting to see the fact that students have this resource, their notebook that contains prior learning and prior experiences so that they can go back to those, refer to those to bring ideas forward and to become more sophisticated with them.

Student: When you figure out the area (?)

Student: ½, ½.

Student: Yeah, make a whole. So that's 5 altogether.

Student: And this is half and another half.

Student: 6.

Student: And it will be 8. So the area will be 8.

Student: How are we going to find the area of this (?).

Student: Well this one looks like 4 times larger than this one.

Student: Okay -

Student: So 4 times -

Student: We know the one in the middle is going to be.

Student: 2 -

Student: Square units, right?

Student: 1, 2.

Student: You can box this, we can put 4 of those in here, so 4 of these, each one is to be worth 2, we've got 4, it's going to be 8.

Student: 2, 4, 6, 8.

Student: This is going to have to be 8 square units.

LB: Connected Mathematics is a complete middle school curriculum, where students are studying units are modules, and each module is intended to build upon mathematics developed in previous units or modules.

One of the things, I think we see in the lesson is students were working at different levels of understanding.

So how did you know like that this had an area of 2?

The launch of the lesson didn't last long. Lisa gave them the basic idea of where she wanted them to go and then gave them an opportunity to really get their teeth into the mathematics, to try out some ideas, to make conjectures, test those conjectures and see where they were going. And by doing that you can see students bringing in different information and working at different levels mathematically. And so part of the beauty is the fact that students are able to bring in their knowledge, test it out, try it and then gain from the knowledge of other people around them.

LB: We've done squares with area of 1, squares with area of 4, and so I wanted you guys to share other squares that you found and which group wants to go first? Tambricia?

Tambricia: We did a square of 4 by 4, and an area of it is 6 square units.

LB: Okay, great job, thank you.

Student: The dimension of our square is 3 x 3 and the area is 9 square units.

LB: How did you find that square that wasn't up right?

Student: We started connecting two dots together.

LB: I think you were able to (?) two (?).

Student: Yes.

LB: And also, how did you find that area?

Student: Because every single one of these triangles is half a square and we added 2 of them to equal 1 square and there is 4. So it's 2 square -

LB: Okay, great job.

Student: We found the area by, first we divided the parts of into little sections and then we just added like another part right here and it equaled 3 and we just halved it and it equaled 1.5. So then we just counted each little section on each side, and it came out to 6. Then we just counted the whole squares in the middle, and then it came up to 10 and that's how we got the area.

LB: Great! Can you also talk about how you found the square in the first place? Because I bet not everybody in here has that square. How many of you guys have that square? Some of you do? Okay, great! That was a pretty tricky one.

Student: How I found it was I tilted the paper over and I was just looking at it because there's other tilted squares so I tilted it and I just saw this where when I tilted the grid paper, I saw the square and then I just drew the lines and then divided it to the inside.

Student: How did y'all get that, because I don't understand. Can you explain it again, please?

Student: I'm going to draw it so it can be kind of easier for you to see. Okay? On my number 3 dots, 1, 2, 3, and that's the 4th dot, 1, 2, 3, 4 and then (here?) and then they go over, then come down and then I drew this line down over here, because all the sides have to be even.

Student: Okay, great, but Guanita, does that answer your question, because you still have like a question mark in your head.

Guanita: No, not really. It's weird. It's just like I don't see how you could just automatically see that, Veronica.

Veronica: All right, which part do you not understand, Guanita?

Guanita: Well, not only do I not understand how you like got that little tilted thing going on, but I don't understand the insides on how y'all did the 1.5's all around. Like, just connecting this part right here, and making it like a whole piece, this little section right here and then we just have to first, the squares equal 3 so we just halved it right it and it equaled 1.5. And all the other ones right here are the same as that one, so they all equal 1.5. So we just added those up. Then it came out to 6. And we just counted the whole pieces right here and it came out to 10.

LB: Guanita, do you see how we're getting the 1.5 now?

Guanita: Yes.

LB: Okay.

Another important part of this process is for students to share with each other, how do you find these tilted squares and that's so valuable that that conversation that students can have with each other.

Student: The area was 8.

And the kinds of things that I'm looking for with students are very unique strategies and also making connections to prior learning.

LB: Okay, so we've had several groups share and, let's see, I'm going to put them all up here and I want to ask you, do we have them all?

So once we've collected all the squares that are possible for a 5 x 5 dot grid, I want to focus students attention on the relationship between the areas that we found and the lengths of the sides. I want to talk about side lengths, and how I can find the length of the side of a square. Now, if I know a side length of a square, like if I gave you a side length of a square, could you tell me its area of even one that's not on here? For example, this square with 9 square units has a side length of 3. And 3 times 3 is 9. But what if I gave you another side length that's not up there, like 5. What would the area of that square be? Kevin?

Kevin: 25.

LB: Yeah, excellent, good job. Okay, how about if I had a side length of 8, Casey?

Casey: 64.

LB: 64. Good, you guys get the idea? Now what if I wanted to go backwards though. And I have areas, because that's what I've got up here, right? I don't have side lengths listed up here anymore. I've just got areas. So what if I'm telling you area, could you go backwards and give me that side length? You think you could? Okay, all right, let's try one. What if the area was 100? 100? Brian?

Brian: That will be 25.

LB: 25, how did you get that?

Brian: Because 25 times 4, a square has 4 equal sides, so I multiplied 25 times 1 is 100.

LB: Okay, what do you guys think? Would it be 25 on one side? I've got some yes's, I've got some no's.

Student: (?) said that if there was one side that was 8 and then it equals 54, you are just timesing 8 by another 8. So if you are timesing 25 by 25, it's not going to be 100.

LB: Brian, what do you think about that?

Brian: That's right.

LB: Can you correct yours? So my original question was, I've got 10 square units, what would the length of one side have to be?

Brian: 10.

LB: Great. Good job. Serisa, you're ready for one? My area is 26 square units, could you give me the length of one side?

Serisa: It will be 6.

LB: And how did you know?

Serisa: Because 6 times 6 is 36.

LB: Great! Okay, Serisa, I'm going to write down what you said? 6 times 6 equals 36, that's, you said that that's how you knew what the side length was, right? So when we have an area and we're trying to figure out, what is the length of the side of a square with that area, we've got to look for, what number can I multiply by itself to give me my area. For example, in these, if I had a side length of 1 on each side, all I had to do to find area was take 1 times 1 and that would give me my area of 1 square unit. And if I have an area of 4 my side lengths are 2, because 2 times 2 is equal to 4. So I've got some new vocabulary for you. I've had a special word that I call that length of one side. Kind of a new word. I call it the square root of the area. So the square root of 4 is equal to 2, because 2 times 2 will give me 4.

Student: Ms. Brown, I have a question. I know that 2 times 2 equals 4, because this is kind of like square roots, but what if you got like 5 or something like that.

LB: Right, and why is 5 a problem for you?

Student: Now, 5 times 5 equals 25, but there's nothing to equal it. Because it's an unequal number, that's why.

LB: So we're going to talk about those kind of cases in a little while and see if we can solve them, see if we can figure them out. Okay? But let's talk a little bit more about the ones that I can do for sure. If I know that an area is 9, you've got that one on your paper, right? What's the length of 1 side, Ernest?

Ernest: It would be 3.

LB: It would be 3. And how do you know it's 3?

Ernest: Because 3 times 3 is 9.

LB: How do you know it's 3 by looking at the picture? Daisy?

Daisy: You can count the squares, like up and down.

LB: Yes, so I can count on the size. So on this one I've got 3 on one side. I can clearly see that that is the length of that side. So this side length is 3. And the length of this other side over here, because this is a square is also 3. I know its area is 9. And again, I'm going to use our new vocabulary so that I can say that the square root of 9 is equal to 3. Because 3 times 3 is equal to 9. Could you guys tell me the square root of 16? Could someone tell me the square root of 16? Okay, so Guanita you are with us. Good.

Guanita: Okay, the square root of 16 is 4.

LB: Why?

Guanita: Because 4 times 4 equals 16.

LB: Okay, and what else can you tell me about the relationship between the 16 and the 4. Dominic do you think you can tie it into the picture. Would you want to come up and show us?

Dominic: I'm not sure, but I think that it's because like if you look at it, it's like 4 going like across and going up as 4.

LB: So the 4 represents -

Dominic: The square root of 16.

LB: Okay, so Guanita, I want to go back to your earlier questions. She brought up a really good point and that is what about these? I think what she said is, I can't find anything to equal up to 5, is that what she said? Well let's take a look at that square that has an area of 5. I'm looking for the square root of 5, which happens to be the length of one side of this thing. What is the length of one side of a square with area 5?

Dominic: Isn't it 2.5?

LB: Why 2.5?

Dominic: Because you have to divide 5 x 2 and 2 x 2 is 4. So you have to add ½.

LB: Okay, let's test it. If I know the length of a side, I can take the side length and multiply it times itself, that will give me the area. Let's test Stephanie's idea. If I take 2.5 times 2.5 will I get 5. Check it out. Serisa says yes. Casey says --- no.

Students: (?).

LB: All right. Serisa you had a question?

Serisa: You multiply?

LB: Yes, because if I've got the length of one side, which Stephanie thinks might be 2.5 -

Stephanie: That's not the right answer.

LB: Okay, so can you guys find it for me? How about if we measure it. Just to get a starting point. You've got rulers in your boxes. You guys can work together on this, it's okay. See if you can figure it out. Try measuring it, and test your ideas.

Student: (?) because look, because I got 2.3.

Student: You start from right here, don't you?

Student: Yes.

Student: Okay, you come right here. At least 2.4.

Student: I don't know. My grade's 2.4 plus 3.

Student: That's what I got.

Student: I don't know.

Student: 2.4.

Student: 7.

LB: You found it? How do you know you got it?

Student: I tried a 25 and I got 5.

LB: Did you get it to come out to exactly 5? What did you get? Let me look.

Student: 5.062, you are so close.

LB: And I've got 5.25.

Student: And this one was 4.9. Okay, so what does that tell you about the number that you are looking for?

LB: (?) 20, 25.

Student: 24.

LB: You're talking about in this part of it? So it's between 2.23 and 2.25? See if you can still find it?

Student: What did you get?

Student: 5.022.

Student: (?).

LB: We're exploring these ideas right now, and these sorts of mistakes that we see students making are all simple areas, so as they refine their understanding they are able to look at their work and say, you know this is what I thought it was at first. I have a better understanding of it now.

Student: Ms. Brown, I did 2.2 x 2.2.

LB: How come? How come you did this though?

Student: Because the 2.3, I measure it first -

Student: 2.3 is in the 5's.

Student: And it was like close to it.

Student: And 2.2 is close, 2.2 is like, it's not enough and then 2.3 goes over it.

Student: It goes over. Okay, what I see a lot of is students taking that 2.2 times 2.2 okay, that's too low, let's try 2.3 times 2.3, that's too high, a lot of them are able to say, okay, well it's got to be in here somewhere. So they are using benchmarks. They are putting a range for, okay, it's got to be in here somewhere and some students, it's hard to get them to stop, because they just keep trying to refine it.

Student: How did it get to 2.23, because I don't know where you got this.

Student: Okay, when we measured it, it was 2.4 and now we've got 2.4 times 2.4, it was too much, it was like 5 and 7 something so we tried 2 because it's like, lower the number down. We didn't try 3, but we've tried 2.2 and then that wasn't, because it was, I think it was less, so we put an extra number at the end. We didn't put a 1 or 2 because that probably wouldn't work so we put a 3.

Student: My (?) is 2.4.

Student: Yeah, but that didn't work, so we have to try something else and then we when we picked the root number, we put the whole number, even with the numbers behind the decimal, we put that and then we got 5.0000002. So that's the closest one. So you got the closest (?). The measurement was 2.4.

Student: I know.

Student: Okay, so then you times 2.23, right, times 2.23, and what's the (?).

Student: We got 4.9729.

Student: Okay, then you multiply 2.23 times what?

Student: Because you put 2.233 times 2.233, and you got 4.9.

Student: I was just seeing. I was just trying to see what, if will get closer if I added another number behind it, but it didn't, because it's still a 9 and an 8. So we took all of the numbers behind the number when we got the root of 5 and we got all the numbers that we multiply that and we got it closer, because it's 5.0000002.

LB: Okay, now, I'd like to talk about this, I'd like for people to share their process about how it is that they first started to tackle this problem, Davie?

Davie: First, I measure it and it was like between 2.3 or 2.2. So I first multiplied 2.2 times 2.2 and it's 4.84. So that's like not enough, so I multiplied 2.3 times 2.3 and it's 5.29, so it's like over it. So it's like between those two numbers, 2.2. or 2.3.

LB: Did some of the rest of you notice that same relationship? Anybody else try it that way? Some of you tried it that way? Okay, and I know, I came around and I heard a lot of other strategies too. I want to talk one thing about Daisy's strategy, where the number she said, she figured out right away, it was somewhere between 2.2 and 2.3, she could even see that by measuring it with her centimeter ruler. It turns out our dot paper, the spaces or the distance between the dots is one centimeter, so I kind of made a little number line here where I've numbered my dots, zero through 9. So I take my square with area 5, I'm going to put it right on my number line, and what do you notice? Alison?

Alison: That the square is going over the number 2?

LB: Anybody notice anything else about it? What do you notice, Daisy, I mean you were the one that kind of started us off on this track?

Daisy: It's like barely going over it, so it's like 2.2 that I measured.

LB: Does it seem like it could be 2.2?

Daisy: Yes, sort of.

LB: Yes, the same that you measured? Okay, Casey(?), how close were you able to get?

Casey: To 6, I forgot, I think million.

LB: Down into the millionth's place, or something like that? Anybody else get closer? Somebody thinks they got closer? Did anybody get it exactly on 5? It's 5.0000000, did anybody get it? Why do you think that is?

Casey: Because 5 is a prime number?

LB: It could be. We're going to talk about this more later. So Syresa I want you to hang onto that idea, because we'll probably be coming back to that, if not today, in a couple of days.

What I'm looking for is exactly 5. Did anybody get it exact? You were able to get it? Guanita?

Guanita: The square root of 5 times the square root of 5 equals 5.

LB: How did you do that?

Guanita: I figured it out in my calculator.

LB: Can you show us that? That's the overhead calculator?

Guanita: I put square root 5 times the square root of 5 and it equals 5.

LB: I'm so glad you brought that up, because Guanita's just discovered a new key on her calculator. Some of you may have noticed it and what do you think that key is called? What do you think it's called? Brian?

Brian: The square root key.

LB: Excellent, good. For now though we're going to leave that behind because we're going to talk about the square root key and how to use it a little bit more either later today or in a couple of days. What I want you to do now though is I want you to use some of the same process that you use on the square with the area of 5 and see if you can find out the side lengths of the rest of the squares on your sheet and label them.

One of the changes that I felt in my classroom when I was a teacher using CMP< and I see, as I watched teachers teach with a curriculum is the environment that's set up in the classroom. Students are communicating their thinking about the mathematical ideas.

Student: 9, 30, 16, 4.

LB: A big part of the success in my classroom is students working together. I have high expectations for them as any teacher would, but I also encouraged them to have high expectations of themselves and each other. They work together to solve problems. My first message to them is you need to take responsibility for your own learning.

I want us to talk a little bit more about how we, our process that we use to find the side lengths of some of these tilted squares, I'm noticing that you guys are having a little trouble with the tilted squares. And Stephanie volunteered to come up and show us her process.

Stephanie: I was trying to find the side length for the one that equaled 8. I took the square root of 8, times the square root of 8 and that gave me 8 with nothing left over. But when I took the square root of 8 again and I took it and I timesed the number he gave me, times the number he gave me again, he gave me one with left over. He gave me 8.0001, which wasn't exactly even like when I took the square root of 8 times the square root of 8.

LB: There were actually more zeroes there than you mentioned.

Stephanie: Right.

LB: Kind of hard to say all those zeroes all at once. So I noticed a lot of you are having that same sort of a problem. I don't understand why it won't work if I take the number that the calculator spits out and timesed it by that same number, it's not coming out. But you know, the deal is here, that what we're trying to find is an approximation and that's really all we've gotten so far. Our calculators can't seem to give us the exact thing that we're looking for here. The square root of 8 is about what? Ashley?

Ashley: I would probably say, 2.82.

LB: Okay, so what I want to do is I want to go back to the number line with that 8 squares units. What we were saying is that if I'm actually measuring the side of a square, then approximation is about all I can get. In fact, if I'm measuring with a centimeter ruler, the nearest 10th of a centimeter is probably good enough, and I can hold this one up to my number line, I've got my 8 square units one that we've been working on here, I'm just going to slide that right up onto this number line and I can see that it's not quite 3, it's really close to 3, in fact, if it was on 3, what would I have?

Student: 9.

LB: A square with the area of 9. If it went all the way over to 3 I'd actually have a square with 9 square units. How about the 10 square units? Where would it fit in this picture? Angela?

Angela: (?) Since it's (?) it can be like bigger and so it's going to be a little more than 3.

LB: Okay great and I can even check it by sliding it up on my number line and I can see that Angela's correct, it is greater than 3, the length of one side of a square with area 10 is greater 3. Okay, so I'm going to use some notation to talk about this approximation and one thing, I've got my square with the area of 8 square units, the square root of 8 is approximately equal and I'm not going to write and equal sign, because I know it's not exactly equal to this, but it's very close to 2.8. So I can say that the square root of 8 is proximately equal to 2.8 units. And the next one I looked at was the square with area of 9 and we know that the square root of 9 is actually equal to 3. The square root of 20 is approximately equal to 3.2 units. And tonight, for your homework, I've got a challenge for you. We're going to talk more about this kind of stuff in the next few days, but tonight I'm going to give you a segment, a different segment you haven't seen yet, and I want you guys to take a look at this segment on page 21, problem number 2. And see if you can apply some of the things that we've learned to date and figure out, what is the length of this side, approximately for that side and how it relates to what we've worked on this afternoon. The second piece of your homework is what I want you to start last now in the last few minutes that we have before we leave today. What I'd like you to start on is a summary for me in your binders. So I want you to summarize for me what you know so far about square root. I don't want you to look in your glossaries, I don't want you to look in a dictionary, I want you to tell me what you think, in your mind, so far about square root. For example, if you had to go home this evening and you had to describe to a younger brother or sister, well we talked about square roots today and here's what I know so far. Go ahead and start on that, if you will please.

The reason why I assigned the homework problem that I did, I'm interested in what they did get out of this lesson. What sorts of pieces from their lesson can they apply to this homework problem? I'm asking them, can you tie in your understanding of square root with the link of this other segment we've never seen before. We'll come back the next day, we'll share our responses and we're going to share, what sorts of things did we write down as we add it to our dictionary. That's information for me. Where am I going to go that next day?

Find the side lengths of some of these tilted squares.

Within the next few days we'll take a look at squares sitting on the legs of triangles and they'll be looking at the Pythagorean theorem. After students are comfortable with the Pythagorean theorem then we come back to this idea of irrational numbers and look at that more closely.

The thing I love about the connected mathematics project is that the students are so enthusiastic about solving whatever the situation is that I've put before them that they are loosing track of time. That's priceless in middle school.

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

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

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