My name is Jim Mamer, I’m a teacher here at Rockway School in Springfield, Ohio and I teach math to kids in grade 6, 7, and 8. This is my fourth year of using the Connect the Mathematics Program. I’ve seen unbelievable growth in my children. I see their level of confidence rising, they believe that they can do mathematics now. There’s never a problem that the children don’t believe that they can solve. Good morning everyone.
Class: Good morning Mr. Mamer.
JM: We’re going to take a look at some of the concepts that we were learning in our last lesson, because the concepts that are going to help you today, if I lay this box out on the table, could somebody in the class come up here and tell us the different dimensions that you see on this box? Okay, Erin, come on up.
Erin: The length, the width and the depth.
JM: And the depth. We’re going to try to get real specific on how we talk about these dimensions. We’re going to start with this bottom part. We’re going to call that the base and on this base how many dimensions are there down there on the base?
Erin: Well there’s only two, the width and the length.
JM: The width and the length. And could you take your finger and run along that so everybody can see where you are pointing.
Erin: This and that.
JM: Okay, that would be your width and your length. And in this dimension here, what are we going to call that?
Erin: The height.
JM: The height, sure, so we’re going to call this the length, we’ll call this the with and we’ll call this the height. Great, thanks Erin. Okay, those dimensions are going to be important today when we do our lesson.
YG: In the unit that Jim is using, in this lesson, the main ideas that we’re looking at are surface area and volume, and what does that mean mathematically? Students are exploring this idea and the idea builds incrementally. The overarching goal of connected mathematics is to build mathematical knowledge and skill with students so that all students can learn more mathematics. CMP really builds a big idea over time.
JM: Last week when we had each group take a rectangular box and design a net or a flat pattern that would fit around the box, we’re going to use that idea a lot today, Dusty and Marybeth. Could the two of you come up here, and Amanda Straight(?) would you come up here and help us, I want to rewrap this net around our rectangular box. So could the three of you come up here and kind of work together to wrap the net around there, please? How did we determine how much paper we were using to cover around that box. What did you guys do to figure out how much paper was going around it?
Student: We measured the length, height and the width. And then we cut it out on grid paper.
JM: You cut it out of grid paper and then what did you start counting on your net there, what things did you count to figure out how much paper we were going to need?
Student: The faces.
JM: Mary Beth, how many faces did you have to count?
Mary Beth: Six.
JM: Could you kind of trace around and show us where those six faces were?
Mary Beth: There’s 1, 2, 3, 4, 5, 6.
JM: Amanda, take one face there and tell us how you figured out how much paper we needed there.
Amanda: Because there’s 10 x 14.
JM: There were 10 x 14 and so when you counted that, you were actually counting the number of what?
Amanda: Squares.
JM: The squares. So your method was to count the squares on each face. And today we’re trying to get efficient methods for finding the amount of material it takes to surround the shape.
JM: The Connect The Mathematics Programs go deep into topics. And the children are going to be using math skills from all different strands. But they are going to go deep into one area.
JM: We’re looking at a rectangular prism here and on this face right here when we’re counting the number of square units that are on this face, what is it that we’re actually counting? Yes, Renee?
Renee: The area?
JM: That’s right. When we decide to look at every face and count the number of squares on every face, including the ends, then we call it a new term. What do we call that then? Eric?
Eric: The surface area?
JM: That’s right. Now, if we decide to try to figure out how many cubes would fit inside of this object, what concept is it that we’re then looking at. July?
July: The volume?
JM: That’s right. That’s called the volume of the prism. What we’re going to be doing today is we’re going to be working for the ABC Toy Company and we’re going to be trying to design packages to house 24 blocks and your job is going to be to go through and try to find packages that will house those 24 blocks. While you do that I’m going to be needing everyone to record their information. The information will be the 3-dimensions of each box that you build. Tell me the volume of each of the prisms that you’d build and I’d also like you to tell me the surface area of each of the buildings that you build. So you have to come up with an organized way, on your paper, that you’d like to organize your data, record your results. I’d like your table to go get a sheet of chart paper and a magic marker, put your information from your four tables onto the big sheet of chart paper. Okay? You’re allowed to use packages of blocks that are back on the back shelf over there and –
The majority of the problems that we work with every day, day in and day out, in the connect the math program, allow the children to have an opportunity to experience first hand with their hands or with collecting their own data and then trying to develop systematic ways of thinking about problems. So every child has that opportunity –
Student: Counting: 18, 19, 20, 21, 22, 23, 24.
Student: And this one works.
Student: Counting – 1 – 24.
Student: It would probably be easier, instead of counting them all out, you know, just to do 4 times 3 to find the area, because then you don’t have to go and count them all.
Student: You wouldn’t have to count this side and then turn it to the other side. Or you could times this side by 2 because this is the same, it has the same side, they can do that to this side, and this side. So it’s 6 times 2 is 12 and then times that by that. There’s more than just those.
Student: Now we have to find the width (?) to put on the graph.
Student: So you would times the width by the height, or the length by the height and you would get the area of this side. And you would do that to each side.
Student: So what about the height?
Student: 4 times 3 is 12. And then 12 times 2 is 24.
Student: There’s 4, not 4, but there’s 6 rows of 4 so you could take 6 times 4 and it would be 24 and that would give you the area of it.
JM: Okay, Amanda, could you try to show me or show all of us how you are finding the surface area?
Amanda: I did 3 times 4 is 12 and then on the other side, 3 times 4 is 12 and then you add 12 and 12 and it gets 24.
JM: Are we done now, if we get the area of that top and the area of the bottom?
Amanda: No, because you have to do the sides.
Student: We have to find the surface area, so we have to find the area of this space and times it by 2 because the two faces are alike. We have to do that to every face.
Student: Okay, the length would be 6, the width would be 2 and the height would be 2. (Because here the width and the length?)
Student: And to get the volume you would do 6 times 2 times 2.
Student: All of them are going to be 24 for the volume, because the number of cubes is 24 for each one.
YG: One of the ways that Connected Mathematics is different from other curricula is they have identified big ideas that are important for middle schoolers in 6th, 7th and 8th grade. They’ve identified important mathematical concepts and they take time to develop those concepts in depth.
Student: Did we find all of the methods? Because we took 6 times 4 times 1 equals 24, 4 times 3 times 2 equals 24. 2 times 2 times 6 equals 24. 24 times 1 times 1 equals 24. 12 times 2 times 1 equals 24. 8 times 3 times 1 equals 24. But like we don’t know if we can find anymore.
JM: You can’t. You don’t know if you can find any more? Ozim(?) what were you doing over there?
Ozim: I was using the multiplication chart to see what was (?) to 24.
JM: How would you know if you’ve got all of them, all of the possibilities? Do you have any ideas on that? Or think about it.
Student: (?) 24, like something times something equals 24.
JM: Okay, so that’s something you guys can work on, okay?
JM: The students learn the mathematics within each lesson at such a high level, that conceptually they are able to move from one lesson to the next and continue to apply what they’ve learned in one lesson to the next lesson. The kids are digging into an idea and working with that idea for a long period of time.
In the State of Ohio we give a math proficiency test to our students in 6th grade and again, in 8th grade. And our 6th grade students, my students, my first couple of years of using the 6th grade proficiency test, their scores were in the high ‘60s and low 70% range. Since using Connected Mathematics, the scores have been 83 to 88% every year.
Student: I really don’t think there’s any more because you could just flip them around and they would stay the same shape.
Student: I don’t know (?) even the most weird shape, it’s still (?).
Student: I know, so on the chart, if you just get it like this and then the next one says like this and then the next one says like this and the next one says like that and the next is like that.
Student: So it works.
Student: How are you going to define the dimensions for that, because I said it was 4 x 3 x 3, but (?)
Student: There’s more than 6 spaces though. There’s more than 6 spaces, you guys. Erin?
Erin: I would say –
Student: There’s a face here then a face here and then a face I here and a face in here.
Student: Because they stack upon each other.
Student: Surface area of 46 – look at that. That’s still less than anything we have already.
JM: When Evan went and built the prism that became almost like a stair step situation and they found out that yes, they could put two of them together and make a rectangular prism for easy stackability, which was one of the criteria we put on a place on at the beginning. I thought that was neat, or it was mathematically correct that they found a way to make these things stack in an order that would work. In the surface area of that step-like prism that he had made, it did have a volume of 24, the surface area was one of the smallest. Type of thinking that Evan demonstrated there, where he was taking the prism and turning it into the step-like prism, and it still fit in any of the criteria that I’ve placed on this problem, that type of thinking occurs a lot in the CMP program.
Student: When we get the volume do we have to times this by this by this?
Student: No, when we do the volume, (?) it’s 24 cubic inches, 1, 2, 3, 4, 5, (?) 24 cubes, Billy? 1, 2, 3, 4, 5, 6, 7, 8 --
Billy: I’ve got 4 more right here.
Student: Okay the surface area for 4 by 1 by 6 is 68. The surface area of 6 by 2 by 2 is 56. Okay. In every package there’s 24 cubes. So the volume has to be 24, because that’s what we’re packaging.
JM: I’d like all of your groups to kind of look at all the charts, think about the information that you calculated on your own to try to see how they are different and how they are the same. If you see anything interesting raise your hand.
JM: One of the values of the connect the mathematics program is the fact that the children are so deeply involved in each problem. The mathematics they learn is much more valuable, I think in this setting because I’m not telling them the mathematics, they are in there digging and trying to figure it out. I still feel I play a very important role though, because I have to make sure that we are moving the in right direction and we are getting the mathematics out of each lesso n. And so I think it takes both of us, it takes the kids really wanting to get it and trying and then it takes me making sure they go down the right path.
JM: Libby?
Libby: All right, everyone has like the same surface areas, just about, like I was in the R group and some other groups had the same numbers and then, let’s see, with that group right there, for B, which has like the most material and which uses the least of the material, that’s exactly what we got, 98 and 52.
JM: Libby, what is that that each of the groups are having the same surface areas in their charts?
Libby: Because they have like the same length, width and height that we used. Like because like they probably had the same shape and then they started the same way we did, but if they were like, we started like this, if they were to start like this it would have been totally different.
JM: What would have been totally different?
Libby: Like, if we would have started like this, the length would have only been 2, but if someone else had it like this it would have been like 12, I think that’s what it is. So that would change what they got for the surface area.
JM: That would change what they got for the surface area. Okay, Jenny, what do you think?
Jenny: I think that she’s wrong because the length and the width can be switched around, but you still have the same surface area.
JM: Okay, and why is that?
Jenny: It’s because like it’s the same shape if you just like turned it around it will be the same shape, just different.
JM: What do you think about Libby?
Libby: What we did at the beginning of class when you showed us the Rice Crispy box, like when you turned it around and like it was still the same thing, so I had this wrong, but so even though could you turn it, like the surface area will stay the same.
JM: Okay. The only thing that’s different there is the way that we’re having the box sit. Okay, the area of the base is what we’re changing. We can turn it all different ways, but what do we know about the surface area of this box? Erin?
Erin: It’s going to be the same thing, you just look at it differently.
JM: Right, if we looked across the charts over there, I think we’ll find, or we can look and see for every box or prism that had the dimensions of 1 x 2 by 12, what do you notice about the surface area of every one of those prisms. David?
David: They all have 72.
JM: Yes, that’s right, they all have a surface area of 72 square units. Class are there other things, other things that you are seeing, other patterns that you noticed on the charts up there? Rene?
Rene: If you multiply it all together you get 36, because 3 times 4 is 12 times 3 is 36.
JM: Good. And how many cubes did we want to have inside of that prism?
Rene: 24.
JM: 24. Okay, Rene, you go ahead and sit down, and take 36, so that is true, does your table see that? Amanda do you guys see that?
Amanda: Yeah!
JM: Had you build that one? Jane, do you remember building that one?
Jane: Yes, that’s the one that kind of goes down like steps, upside down steps.
JM: Now, does that work? There’s a 3 x 4 x 3. Do those dimensions work?
Amanda: When we found it it did.
JM: Could you build that so we can take a look at that. Evan?
Evan : We thought about the same kind of things. I think this is what they did over there. It’s – it is 3 x 3 by 4, but it has some of the blocks cut out. It’s not really a rectangular prism so it really can’t be categorized with all the others.
JM: Was that the kind of prism you guys have built, Jane?
Jane: Yes.
JM: What other kinds of patterns were you seeing when you look up here. Evan?
Evan : All of the dimensions of the different ways that you could do it were factors of 24.
JM: Can you go up there and explain that or show that?
Evan : Like 12, 24, 8, 6, 2, 3, and 4, they are all factors of 24. Like 12 divided by 1 is 1, 12 divided by – 24 divided by 1 is 1, 24 divided by 12 is 2, 24 divided by 8 is 3, and –
JM: So you were looking down the columns and you saw that those numbers were all factors of 24.
Evan : Yes.
YG: We could see students making connections back to prior knowledge. They were talking about area and that was a skill and concept that they had developed earlier. They were talking about factors and how factors were important to them, to find all of the possible combinations. So we get a glimmer that students are bringing prior knowledge and using past experiences to help them.
JM: Jonathan, your table, you guys had a method for finding the surface area, where you wrote down X plus Y plus Z would equal the surface area. I’d like your group to come up here and show us mathematically how you were getting that.
Jonathan: Those were kind of like the width times the length times the height, and we use like certain symbols for them(?) and that’s what basically that is. And also they also represent the top times 2 and the bottom times 2.
JM: You get the top times 2?
Jonathan: Actually the front, the top and the right times 2.
JM: Why are you always multiplying this times 2?
Jonathan: Because there’s 2 of them, so --
JM: 2 faces? There’s an example of a prism right down there on a table beside you. Why don’t you pick that up and like kind of show us what you’re saying there again.
Jonathan: Since there’s, here’s the top, we did that times 2 since there’s one on the base and then for the front we can times that by 2 because there’s the same one on the back and for the right it’s the same thing on – that’s why we times it by 2.
JM: Great. Thank you. Did any other groups use a method that was different than that to get the surface area. Grace, your group did? Could you guys come up and maybe, beside their group you could write down how you folks did it.
Grace: Okay, what we did is we found the area of this face and that was 6 and then we found the area of half and the top was 12. And then that was 6 also and that was 24. And then since we knew that we had found the area of half, and then we knew that this was like the other half, so then we did 24 plus 24.
JM: Okay, Julie? Go ahead Julie, go ahead and say what you wanted to say.
Julie: One of these guys wants to be 8 instead of 6.
JM: Great. That’s great.
Julie: Then the surface area would be 52.
JM: Great. Okay, Abby, Patrick, I’d like somebody from your group to come up to the chalkboard and show your method for how you folks were finding the surface area and let’s go ahead and use this prism right here. So could somebody come up and write down how you found the surface area of that prism and you can write it on the chalkboard.
Allison: What we did, we found the area of all the faces, for this face it’s 12. We did 12 plus, for this face on the side here, which is 6, and then we found the area for this side which his 8, and then we found the area for this side, which is also 8 and then we also did it with the other side, which is 6. And this side is also 12 and we added these together and it’s 52.
JM: Okay, great, Allison. Okay, so on this one here, let’s say that I had the prism 6 sitting this way, and I need your help. This 12 right here, where does that number 12 come in? Erin Carter?
Erin: The right side?
JM: Okay, Erin, after we had that unit of 12 square units on the right hand side, where could we get an area of 6 from on this group?
Erin: The top one?
JM: Okay, so we could say it was the right plus the top side, plus the area of the 8th, where was one of the 8s?
Erin: Front.
JM: Okay, plus the front side, plus another 8. Where would we get another area of 8? Jenny?
Jenny: The back side?
JM: And then Erin, where would we get another 12?
Erin: The left.
JM: The left hand side. The left hand side and then – and I’ve missed one of the sides. Where do we get the area of 6 from Erin?
Erin: Bottom.
JM: That’s right. Okay, what I’m going to do now is I’m going to put all three of those methods over here for us and I want you to write down each of the three methods in your notes and these three methods are going to help us find the surface area of rectangular prisms. Okay, we have our first one out there, the first method written down. The second one we’re going to write down will be this method right here, we’ll call this method number 2 and on that method we said we were going to multiply 2 times the quantity, top area, plus the left side area, plus the front side area, we’re going to add those together, get the sum and multiply that quantity times 2. That was method number 2 that I need you to write down. And then the third method we came up with was the method that Osim’s group came up with which is right over here where they added the different faces and that was, we added the right side, Osim can you help me out over there, it was the right plus the top plus – okay?
JM: Okay, I’ve taken those three expressions and rewritten them over here, and I’d like you to put those in your notes, and at your tables, I want your tables to discuss, what do you similar about these three expressions, they way that they get the area, or the surface area of the prism, what’s similar about the three and maybe what’s different about all three of them. Have a discussion in your table about that, once you get these three things written down.
YG: When I talk with other teachers and as I go around, because I have the opportunity to work with teachers all around the nation, you hear the same thing. They are working harder than they’ve ever worked before, because the curriculum pushes the teacher as hard or harder than it pushes the students. But at the same time they are more enthusiastic, because they are learning more about their students and they see their students growing mathematically and seeing their students able to understand and retaining ideas as they move forward.
Student: Once you try each of these methods then that will give you the surface area.
Student: Okay.
Student: (?) is that just like half of it (?).
Student: Yes, what they did was they found half of it and then they did that times 2 and they got their surface area.
Student: I think number 2 would be the quickest area too to figure it out.
Student: Yes, but I think number one is definitely the easiest.
Student: So number 1 would be the easiest and number 2 would be the quickest.
Student: Number 3 is always the hardest.
Student: (?) it would be harder than the other one just because you have to add every side all by themselves.
Student: There’s not very much of a difference, because in all of them you are giving the area of all the faces and then you are adding all of them up to get the surface area. It’s just done in a different way.
Student: The way 2 and the way 3 go good together because way 2 is just that we had top plus left plus bottom, it’s half (?) instead of doing each one individually.
Student: Number 1, they took 1 and timesed it by 2 and did that for all the sides.
Student: (?) number 2 is going to be easiest, because you already know 2 times, so you take the top times 2, there’s a top and bottom
or all you have to do is add the top plus left plus front and then times it by 2. Yeah, because that’s like both sides of it.
JM: Okay, class have you found anything interesting about any of the three expressions, anything you can share with us? Jane?
Jane: I think that it would have been much easier if you did either 1 or 2, especially 2 because number 3 you have to add all of the different numbers up. And if you just used number 2 you had to do like 2 short steps. You add up the top left and front and then times it by 2.
Student: We didn’t think that you should use number 3, because you have to do a lot more than 2 steps to get your final answer and it’s kind of confusing.
JM: Kind of confusing? Okay, did method 3 work?
Students: Yes.
JM: Yes, absolutely. This would give us the surface area, we could always use it. Is it the most efficient way? Is it the fastest way that we could do it? Jenny?
Jenny: No, because you have to add the tops, the bottom, the left, the right and front and back and you have to count each one of those squares? It would be like take most your time up to do that.
JM: What we’re going to do now is we’re going to try to figure out a way that we can organize all the data that you folks collected on these charts. Each of the groups have found about 6 different ways to house these 24 cubes, we’re going to –
JM: At this point in the lesson I felt pretty comfortable that most of the children had a pretty good understanding that there were ways that were more efficient than others to find the surface area of rectangular prisms and so at that point I wanted to go after one more big idea from this lesson. And that was the idea that, as the rectangular prisms because more and more convinced or more cube like, the surface area continued to get smaller and smaller.
So at that point I told the children that we were going to walk over the library where we already had a bunch of boxes laid out and the prisms that were built there look just like the ones that they had built earlier, but we had arranged them in an order that went from long and skinny with a great surface area to a prism that was very cube like even though it wasn’t a total cube and we had already laid out what the length, width and height was for each one of those prisms.
We had also said that the volume was 24 in all of those, and then we had the surface area going from the higher surface area all the way down to the lowest surface area, which they have on their chart, but their charts just weren’t set up in a manner where they could see that happening.
We gave them 5 or 6 minutes at a time to look there, and the discussions were really nice and interesting and a lot of kids were seeing some of the things that we were hoping they would see.
Student: Okay, the second one, you see the width, it has (it as?) 2 and ours is 1, so that one’s different.
Student: They have all of the heights 1, and this (?)
Student: (?)
Student: Yes, they do, they switched it –
Student: Well down here is 2 and 2.
Student: Okay, here we have these two switched too.
Student: Yeah, all the ones (?) the ones that are –
Student: You can basically start it and have 1 on the base, and then it has 2 and then 3 and then 4 and they stack it up and also that the surface area gets smaller and the height gets bigger so –
Student: Does anything happen with the width?
JM: Okay, kids, I want you to go back to the beginning of this lesson. What I want you to think about now is if you are the packaging engineer, we walked over the library and we looked at a different way to organize all those boxes that could have been made. I want you to think about those boxes that were over there, all the different packages. Which of those packages, which package would you choose, if it was your job to make the box, for the company, to house those 24 cubes? We want you to have that discussion with your group, within your group right now.
Student: We picked the 3 x 2 by 4 because it requires the least amount of paper to build, so you would have, it would be the cheapest one. It’s not, it won’t be a very big box, because like the 24 x 24 x 1 box, that box would be like that big and this one is only going to be about that big or something.
Student: (?) get that (?) cubic blocks that you can just sit on top of each other.
Student: And the shape (?) and more compact, the surface area got lesser and lesser. Like when it was like the 1 x 24 by 1, you said this area was 98, but with the 2 x 4 x 3, it (?) at 52.
Student: Number 2 is the 3 x 4 x 2, isn’t it? Okay.
Student: Which is this one right here?
Student: It says it’s the smallest one there, it will hold probably a little bit less.
Student: The same as all the other ones, but it just – it’s smaller –
Student: It takes less (????)
Student: It’s definitely number 2.
JM: Okay, when we came back from the library and you saw the chart the way we laid it out there for you and we had all the cubes sitting there, all the rectangular prisms, when we looked at that chart, what was a pattern that you noticed, Abby?
Abby: The more compact it got, like the smaller the surface area was.
JM: Okay, Chris, so what did you notice?
Chris: It was more or less like a square, a cube.
JM: It became more or more square-like or cube like and what happened to the surface area as it got more and more cube like?
Chris: It increased.
JM: The surface area increased the more it became like a (?).
Chris: It decreased, I mean.
JM: It decreased. Okay, any other thoughts on that, or any other patterns anybody noticed? Grace?
Grace: I noticed that as it became more like a cube, the height increased.
JM: As it became more like a cube, the height got taller and taller, that’s neat. Anybody have any idea why that happened? Libby?
Libby: Because when the cubes like get smaller they all like hook together, it gets taller, because when you like compact them it just makes it like that. And the surface area decreases, because like when they are like all connected all the sides that are like facing out, there aren’t as many because they are like hooked together, but when they are in a straight line there’s only 1 side that’s actually touching with the rest of them, but when they are together there’s like 2 and 3 sides.
JM: When they are together like that there’s all those faces on the inside that never see the outside.
Student: Like this one (?) but in a straight line there’s at least two or three showing.
Student: I was going to say like for each cube in the straight line one, only 2 faces of that cube are not showing.
JM: That’s neat.
Student: But if you pack it together like this, the least faces that any one cube has not showing is 3 and that’s the corner one. All the others are more than 3 that aren’t showing.
JM: Great! These 3 out on a corner piece, on the corner piece we have those 1, 2, 3, faces showing, 3 faces are inside. On a side piece, you’ve got 1, 2, faces on the outside and then you’ve got how many on the inside?
Student: 4.
JM: 4, we lost them, right? And that’s how that surface area gets smaller and smaller, that’s awesome.
JM: I would have never guessed that Evan and Libby would have found that as the structures became more and more cube like that more and more the faces on the inside because hidden, that totally blew my mind away when they did that. And as they reasoned that, it was like, wow! You know, I was thinking to myself, yeah, that makes sense. I’ve been doing this for years and years. I never really totally thought about that method and when Evan brought up the point of if you have the long narrow, the 24 x 1 x 1 prism, when he started talking about, you are just going to have 2 faces, that are covered or hidden, except for at the very end, the last two cubes will just have one face hidden, that’s some good thinking going on there.
JM: Okay, you’ve done a great job looking at the surface areas of all these different prisms and we found, you found that the more compact the prism became, the lower the surface area became. My question to you is, do you think this would work for any number of cubes? I’d like you to go home and think about, if these 36 cubes, what would be the best way to package it to save money. And then tomorrow, when we come in, we’re going to be digging hard into this again, we’re going to start doing several different prisms, with several different amounts of cubes. You guys did a great job. Thank you everybody.
JM: The best thing about teaching for me is opening doors for children to believe they can learn anything they want to learn and that’s any subject and it’s life. I think that all kids need to believe that they can learn and if they work hard there is a way to find out information. And I think too often we shut doors to children and I’m going to open those doors for kids.
Major funding for Modeling Middle School Mathematics is provided by the National Science Foundation.