Common core State Standards
- Math: Math
- 7: Grade 7
- G: Geometry
- B: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Transcript for Discovering Surface Area of a Cylinder with Chris McCloud -- 7th Grade Math
Chris McCloud: Hi my name
Transcript for Discovering Surface Area of a Cylinder with Chris McCloud -- 7th Grade Math
Chris McCloud: Hi my name is Chris McCloud, I’m a seventh-grade math teacher here at School of the Future
CM: Today’s lesson is an exploratory lesson, or a constructivist approach, where students are trying to find the equation for surface area of a cylinder on their own without being told in a direct fashion by me. So it’s not me saying the formula is this, use it, it’s them finding it on their own and then using it.
CM: To me, today’s lesson is not like the mechanics of the surface area formula, like it’s not about me checking if they can multiply with decimals, that’s not what I’m looking at. I’m asking my students to grapple with these high-level ideas, the big ideas.
Voice Over: Chris is teaching kids with mixed abilities as well as special needs kids with paraprofessionals in the room. Chris began today’s class by having the students complete a worksheet that tests their knowledge of two mathematic formulas for circumference. We join him ten minutes into the class, after the students have completed their worksheets.
CM: All right, let’s put our pencils down, stop where you are, that’s where you need to be. And let’s bring it back up to the front, so, calculators down. All right, so we’re gonna go off to the cards, we’re gonna talk through number one. Two of those expressions measure the exact same thing. Which two are they, nine of clubs.
Student: Um, pi times diameter –
CM: So you’re saying pi times diameter, and what?
Student: And two times pi (r).
CM: And two pi (r). Okay good, you are correct. You’ve identified the two, now let’s talk about why. Seven of hearts, which parts of these two expressions are exactly the same, nothing’s different about them. Seven of hearts, who are you? Which part of these two parts are exactly the same? Just look at them, what’s the same?
Student: Um, pi?
CM: The pi part, exactly. So these pi parts that I just underlined in red are just the same, good. Now, moving on, seven of clubs, tell me how, tell me how this green d and the two and r are related. Seven of clubs, who are you?
CM: Oops, sorry, pass. Six of spades? Dumel, how is the d related to the two and the r? What does the r stand for?
CM: Okay, how can you relate a radius to a diameter?
Student: Radius is the whole thing but diameter is half.
CM: Ooh…other way.
Student: Diameter –
Student: The whole thing.
CM: The whole thing. That makes the radius half. So okay, work with that. Why does 2r then make sense, two times r, why would that make sense?
Student: Two times r is the diameter.
CM: Ooh, there you go. Two times r equals your d. So these are the same formulas, the pi parts are the same, and two times the radius is just your d. Okay, good. Um, moving on, let’s have the next student come up, put your work up for number two, your circumfrence and area calculation. This will be…four of diamonds. Let’s go Mr. Abuyan! The rest of us, make sure your pencils are down. Kay-Wen, respect, let’s go. Shh. Just walk through it, so let’s start here, I like what I’m seeing so far, okay. You can sit down, you can take a seat. What do we like about this? What do we like about Adrian’s work up there? Let’s just point out the things we like. What do we like, Trevon, start it off.
Student: Um, how he has the equation.
CM: Exactly, how he has the equation so every calculation starts with his equation. I like that too, good. Kay-Wen, keep it moving.
Student: He does it step by step.
CM: What do you mean?
Student: Well, for the circumference, see how he switched the d –
CM: Into a…?
Student: Where is it?
CM: Here. Into the diameter of 60, so that’s one step, that step’s called substitution. Very good, I like that, Kay-Wen. What else do we like, Corrine?
Student: I like how it’s organized and that there’s no mistakes and he’s not skipping steps.
CM: Absolutely, did any of you guys have to do a double take, like look at it like ugh, what does that say? No, it’s very neat and organized, it’s clearly stated, very good. Taylor, what else?
Student: Um, his answers are circled so that we know where to look for the actual answer.
CM: Answers are circled, this one is, you can circle that one. I see something else, the dot your I’s, cross your T’s. What am I talking about, Jonah?
Student: He has the labels, he has feet squared.
CM: Yeah, he has feet squared for area and feet for circumference. Now, real quick, what’s the difference between circumference and area? What is the difference? Go ahead, Brandon.
Student: Circumference is almost like a perimeter of the outside of the circle –
CM: Perimeter of the outside of the circle, or like a distance around the circle, very good, and that makes the area what?
Student: The area’s the inside.
CM: The space inside, very good, so if you colored in the circle that represents your area. Awesome. Okay I would like at this point, everyone flip your page. Flip your page. You’ve got two minutes to start off a think tank, but before you do look up here real quick. So some of you guys may or may not remember the snap bracelet. But it’s this like flat to start off and I can snap it on my wrist. Now I have like a cool looking bracelet. Now what is, if I ask you in that question, I say that the maximum circumference of a snap bracelet is eight pi centimeters, what do I mean? If the maximum circumference is eight pi centimeters, what do I mean? Go ahead Dave.
Student: Maybe you mean like it can only go up to um, you said eight?
CM: Eight pi centimeters.
Student: Eight pi centimeters, and it can’t go above.
CM: Absolutely, it can’t be bigger than that. The maximum circumference or distance around this snap bracelet, eight pi centimeters. Now obviously that’s a little big for my wrist, so think like NFL offensive lineman or something like that, some big dude. He’s gonna have – this would fill out this snap bracelet. Good, use that idea to formulate your opinion on that think tank. Two minutes, silent, go. True or false but defend it either way, tell me why.
CM: Alright now stop, stop where you are and move into a be heard discussion with one other person in your table, talk this out.
All students talking.
CM: Good, that’s a good observation.
CM: All right, T to me, bring it back, bring it back. There’s enough discussion out there, I think we have the ideas. So for those – let’s do a recap. The statement was this. If the maximum circumference of a snap bracelet is eight pi centimeters, then when I straighten it out, the length of the bracelet is eight pi centimeters. True or false? Okay, raise your hand if you think it’s false. Anyone think it’s false? Not one person? Good, so the rest of you then think it’s true, let’s talk out why. Why is this true. Let’s start here, Grace, start it off.
Student: Well one example is a piece of paper. Say it’s eight inches long, if you make it a circle, it’s still eight inches long.
CM: Agreed. Moving, keep it moving. Keep it moving Naji, step in.
Student: I think it’s true because you can’t change the amount of the circle.
CM: Good. Naji’s getting at the fact it’s like, this, did I change anything in the bracelet? Did I cut off any pieces? Did I extend the bracelet? Or did I just unfold it? Okay, okay, okay, who haven’t I heard? Taylor, go ahead.
Student: Okay um, when you put the snap bracelet on, um, the circumference we originally see as an edge, and when you straighten it out the edge is still gonna be the same thing no matter what wrist it’s on.
CM: Ooh, you guys got it. That’s good, that’s good. Adrian, last thing.
Student: The circumference, when you measure it, is like you unrolled the circle and measured how long that line was.
CM: Good. Hang onto that idea. So what I would like you to do, take your think tank paper, leave it think tank side up, and put it on the corner of your desk. This is your one and only hint from your teacher for this period. So leave your think tank paper out, think tank side up, you’re gonna need it some time in this lesson. You’ll be like “ugh” and I’ll be like “look at your think tank.” The rest of your stuff can go away.
CM: so here’s the scenario. Uh, the supermarket around the corner used to sell Grandma’s favorite mandarin oranges. For reasons unknown, they no longer do, and it really upsets Grandma. Jackson, a thoughtful young seventh grader, remembers this fact as he’s thinking of what to get his grandmother for her eightieth birthday party. Cans of geisha mandarin oranges, of course, right here. So the problem is this – there is one potential problem. Jackson has two cans to wrap but only has 550 centimeters squared of wrapping paper. Will he have enough paper to wrap the cans? Take a look, stop what you’re doing, eyes up. Two cans, one is smaller than the other. They are not the same can, okay? Look, one’s bigger. So he bought these both for his grandmother, nice guy, and he’s gonna see if he has enough wrapping paper to put around them and wrap them both individually. Okay, go ahead, before you start working at your table group, I want you to spend one minute, I’m eying the clock, and I want you to think of your own strategy. What might you be able to do, what do you need to know, how will you find it? Get something down in the strategy box.
Students start talking.
CM: Independently, time out, shhh. Corrine, Kay-Wen? It’s the independent write. You get something down on your own in that strategy box, then you’ll talk.
Student: Can we explain it or do we have to show you?
CM: No, you’re just gonna get a strategy down. What do you need to know, what are you trying to find out, what does the problem require of you?
CM: Look up, if you’ll notice each of these labels is scotch taped on. You can feel free to remove the scotch tape. In fact, that’s kind of a hint, you should remove the scotch tape. Just don’t rip the label off, I’d like to use them again. So please don’t just rip the label off, you can definitely take it off, in fact you should. Go ahead, that’s it, the rest of you guys. This is a country point challenge, the first three tables with the correct surface area, the correct strategy, 25 points each. Go!
Student: We’re using a basic strategy, but I think it might work.
CM: Let me, let me see. First of all, start your strategy off, what was your strategy?
Student: Our strategy was um, height by circumference would equal the surface area of the cylinder and the reason is because –
CM: The whole cylinder? Which part?
Student: This part.
CM: Which part?
Student: The part around.
CM: Okay, is that it?
Student: And then you have to add the top circumference and then you’re done.
CM: Top circumference?
Student: Top and bottom area, shoot!
CM: What are we finding?
Student: Yeah, we’re finding…yeah!
CM: I agree with some of what you said. What you said about the middle piece, I’m not gonna argue with it, it’s correct. But there’s something else you’re missing. What else is part of it?
Student: The area.
CM: Surface area asks us to do what, Damel? What is the surface area?
Student: You need the top.
CM: You need the top, you’re right. How many – is it just one top, or is there two?
CM: Okay, so you need two tops and that middle piece you just talked about.
CM: Did you guys get a strategy yet?
Student: Yeah, we did. We did, um, this.
Student: Pi (r) are just the same so we just doubled it.
CM: Pi r squared…you doubled what?
Student: We doubled seven, because they’re the same, one, two, so we find –
CM: Good. So you got most of it, what else do you need though?
Student: We need to find this area.
CM: Okay, why am I talking to you, you’re right. Keep working. You guys are ready to be assessed?
CM: Okay, tell me how you got there, more importantly.
Student: Can I tell you?
CM: Why don’t we start it off – you start it off, go ahead.
Student: So first we found the area of this, which is –
CM: How’d you do that?
Student: Length times width because it’s two-dimensional.
CM: Okay, cool.
Student: And then we got 21 times seven.
CM: Hold on, let me stop right there. Pick it up, Corrine.
Student: Okay then we did, we found the length and the width of this and we got 21 and seven.
CM: Yeah, you got that – we’re here. Tell me about the next piece.
Student: So, um – area equals pi r squared, so we had to find the radius of this to get the whole idea.
Student: So we did area equals pi and then three times –
CM: The only thing I’m gonna say real quick, how many of these circles are there, Jasmina?
CM: Did you guys multiply by two?
CM: Fix it, hurry up, hurry up, hurry up!
CM: All right! I wanna ask everyone, stop where you are, that’s where you need to be. Put your pencils down. Let’s debrief this a little bit. Um, all right, so, three of clubs, start it off, talk to me a little bit about what your initial strategies were. What did you do to start off?
Student: Our initial strategy was to find the area of the label which was –
CM: Hold on one second, you say area of the label like, I’m looking at a can, what did you do with the label?
Student: We took it off.
CM: And what did it look like?
Student: It looked like a rectangle.
CM: It certainly did, it certainly did. It looked like a rectangle. Then, keep going.
Student: Then we did length times width.
CM: Okay, so you had a length here and a width there, and you found the area of the rectangle. I agree with you. Good. Let’s pick it up with someone new. Nine of hearts, jump in.
Student: Okay, so after that we found our length and the length was –
CM: Let’s leave the calculations because some of these cans are slightly different so they won’t make sense for each table group. Let’s just ignore – the idea is this, length times width of your label. And then what next?
Student: We moved onto the base of the can.
CM: Which was what type of shape?
Student: Um, a circle.
CM: A circle! So if we’re calculating surface area, we have to add in those circles, and how many were there Trevon?
CM: There were two. So I’m gonna put a times two here.
Student: So and then we –
CM: Stop right there. Stop right there. So so far you guys are developing this formula that starts off as surface area equals, looks like length times width, and then you have a plus, and then, three of spades, how do we find area of circles, I forgot. How do we find area of circles?
Student: We use circumference formula?
CM: Circumference formula? For area of circles? That’s all right, you have it right, so you just –
Student: Yeah, it’s pi (r), exponent two.
CM: Pi (r) exponent two, or pi (r) squared. And Veronica, how many circles were there again, I forgot.
CM: Two! So, good. So we’re gonna have this two out in front and, good. Nine of – eight of diamonds, why did I just put a green two here? Eight of diamonds? Why did I just put a green two here?
Student: Because you multiplied both bases of the area of the circle.
CM: You’re finding – okay, we’ll work with that. We’re finding the area of both bases, the top and the bottom. So that’s two areas. Okay, good.
CM: Okay so so far so good, I like what I see but now things are just about to get a little tricky.
CM: What we’re gonna do next, I’m gonna take your – take the can that you just calculated the surface area with, I’m gonna trade it with someone who has a different size can. And you’re gonna flip your page and you’re gonna do the exact same thing, except what? There’s no label!
CM: You can figure this out, there is a way, I promise you. You will find it. Get it, go!
CM: When you guys did your think tank and we had this eight pi centimeters – eight pi centimeters circumference and I unwrapped it, what was the length of that?
Student: It was the same length.
CM: The same, eight pi, right? Eight pi centimeters. So, same idea. If the label we don’t have when we unwrapped it, if we know the circumference when we unwrap it, it would still be what?
Student: The same.
CM: The same, right? So even though you don’t have the label, you know that the circumference when it unwraps is your length. Can you find circumference somehow? Do you remember how to do it? Look back at your do now. Look at your do now. Use Jonathan’s, oh it’s right here. Okay, you don’t have it filled out, but it’s one of these two.
CM: So what’s happening? Do you guys understand the strategy going on right now?
Student: I’m kind of confused.
CM: Can you think back to your think tank when we had that bracelet and we unwound – it was eight pi centimeters, right? The circumference? Wh thappened when we unwound the bracelet, the length of that was what, right? Eight pi centimeters. So same idea. What’s gonna happen, you have your label – your label represents what measurement of a circle, or the cylinder? When it’s on? You got it, circumference. And when you unwrap that circumference it’s still the circumference – or, unwrap the label, it’s still the circumference, right? Well we just don’t have a label but you know that when you unwrap it that length is your circumference. Can you find circumference, is there a way you can do it? What could you do?
Student: We already did it, pi – pi to the seventh.
Student: We found the diameter and then we times it by 3.14 and we got 21.98 and we round it to 22, times eight equals 176, and –
CM: So okay, you got 22 for your circumference? That’s good.
CM: Now how are you gonna use that to find the area now of this piece? Remember the shape.
Student: We’re gonna times it by eight.
CM: What’s eight?
Student: Eight is the width.
CM: Show me.
Student: It’s this.
CM: Good, you should do it.
CM: Right here. Sweat it. Okay, let me hear it – strategy.
Student: Well, we did the same one earlier because we had to find the area first, of this.
CM: Yep, okay, you did that.
Student: Then we found the area of –
Student: Which was eight.
CM: How did you find the area – I agree with you. So, Kay-Wen, you told me you found the area of this and you probably multiplied it by 2 because there’s 2. Now how did you find the area of this middle piece?
Student: We measured this with a rubber band.
CM: Okay, that’s creative.
Student: And it equals exactly eight.
CM: Interesting, I like the strategy. Okay, you guys are in for 50.
CM: You guys, where I left you, is you had found the circumference of your label. Picture this as your label, you found the circumference or length, how did you find the area of the whole –
Student: Well I think what we did is we backtracked a little from yesterday and we had to find the height of it to complete the problem.
CM: Got it, got it – you guys are in. Okay!
CM: First thing, I just want to say that was excellent enthusiasm, I absolutely loved it. You guys were all into it and all trying your absolute best, which is all I can ask for as your teacher. Next thing I’m gonna ask is make sure you put your rulers down, pencils down, all distractions away from you. Now we just need pure mental focus for the next five minutes. So don’t worry about the things that are gonna go up on the board, I have everything – I’m gonna take this pressure off of you, I have everything written out on note sheets for you guys after this lesson. So right now all I’m asking from you is 1000% focus, okay, with your pencils and everything down, okay here we go. So when you were calculating the surface area of your second cylinder, five of spades, did you do anything differently when you tried to find the area of your two bases. Did you do anything differently? Is this you? Five of spades, who’s this?
Student: I measured – it was centimeters and –
CM: Hold on a second, Naji, you’re gonna tell me some numbers, that’s okay. I’m just talking about when you found the area of your circles, the bases, like the top and bottom of the can, Naji, look here. The top and the bottom of your can, did you do anything differently?
Student: I multiplied that by pi.
CM: Pi…what’s your formula for that?
Student: Pi times d.
CM: That’s circumference, we’re talking about area of those circles. You need a bailout?
CM: Okay, go ahead, pick one.
Student: Two pi r.
CM: Two pi r…
CM: Squared. So the idea here is that when you guys found the areas of the circles on the top and the bottom, your bases, you didn’t do anything differently. It’s the same – that was the same. That didn’t change, that wasn’t the trick. That wasn’t the hard part to understand, so good. So if I’m gonna rewrite this formula, I’m not gonna change that part either, you already found out a good piece of the formula. That 2 pi r, 2 pi r squared piece, that sticks. That’s good. That’s good. All right now the trick though was this next piece with the label, because the label wasn’t there. So what we want to hear now, six of – six of hearts, where are you?
Student: Not here.
CM: Seven of diamonds?
Student: That’s me.
CM: Okay, Carly. What am I tracing out right now?
Student: You’re tracing out the circumference of the base.
CM: The circumference of the base, which we know as 2 pi r or pi d at this point. Good. So that’s cool if you had the label, like that would be really neat if you – So basically that’s what your label does, is it traces out your circumference. But did you have a label this time?
CM: No, you didn’t. So what was the trick, what did you guys do to figure out how to move around that?
Student: We measured the diameter and found the circumference which is –
CM: You measure the diameter of your can and multiply it by pi to find your circumference?
CM: Which on this label diagram would represent which piece, the w or the l?
Student: The length.
CM: The length. Does everyone see that, she’s saying even though the label’s not there, so what? I know that this label was wrapped around the can and it represents your circumference just like our snap bracelet did in the beginning of class. Like when you unfold it, it’s still the distance we’re talking about. It’s still that distance. So we can then kick out the l, we don’t need the l anymore, and replace it with what? Circumference as Carly just said. So she wants to put now, circumference here, which we could say is what, 2 pi r I’m gonna say, one of our expressions for circumference. Good. Nice. Let’s keep it moving.
CM: All right. We’re almost there, bring it back, stay with it. There’s one last piece we need. I was on four of hearts, who are you? Ally, what’s the last piece we need?
Student: The last piece would be height because we have to replace the w.
CM: Yeah, because we don’t know, we didn’t have the height, we just had the can. But you could have easily tooken your ruler, the height of the can is whatever it is, so this is our H here, the height, which you can replace with – that was the side of your label. And now, now it’s starting to make more sense. We have the area we need for the base, top and bottom, but there’s one last calculation that needs to be made. Eight of spades, what is it?
Student: Um, the last calculation you need to make is, um…
CM: We were right here, we had l times w for area, we replaced it with some stuff. But we can still do the same kind of –
Student: You need to find the length times width.
CM: Length times width of our label, that wasn’t there, so instead we’re using circumference and h so put them together. When you multiply those together, you would get…?
Student: You multiply the area?
CM: To find area, we multiply – it’s okay, Dave, you’re good. You want to pass it on?
CM: Go ahead.
Student: I used this – pi times diameter times height.
CM: So, Brandon, pi times diameter which he’s using pi d, that part, I’m gona use the 2 pi r just to show this. So same idea, you’re multiplying your circumferene here times your height. So instead we would have 2 pi r for our length and our width then, height. So then that’s kind of messy, I’m gonna rewrite it one line below. Surface area equals 2 pi r h times 2 pi r squared. Thank you, you just made my day pretty easy. What’s up Trevon, go ahead.
Student: How come on the top you have it spaced out in parentheses?
CM: Right here?
CM: Because when you multiply, Trevon, does it matter if you do one times two times two times two or the other way around? Does it matter which order you multiply things? Like two times one is the same as one times two, right? So same idea. We can do two times pi times r times h in any order. It’s fine, we can remove parentheses like that. Any other questions here? Go ahead Kay-Wen.
Student: Now we know a surface area but how do we find the volume?
CM: Ooh, that sounds like your segueing into tomorrow’s lesson, let’s wait. Taylor?
Student: Can we just say that, um, to find a label you can two 2 h times c? I mean 2 c times h?
CM: 2 c times h. Hold on a second. I like what you’re trying to go, but 2 pi r represents a what? 2 pi r is…?
Student: Radius. I mean, circumference.
CM: Is a C. So what you want to do, what you’re trying to do is do a C times h, so you can do that, circumference times height. Well, you’re gonna want to use this one because chances are I’m gonna give you a value for r or diameter, which is just easier to plop into this one.
CM: Okay, good. Uh, real quick, last thing before we go today, does everyone see at the bottom of their page, Mr. Absent Kid is there? So absent kids is gonna show up tomorrow and they’re gonna be like ugh, I’m kind of confused with this formula, Chris. I understand 2 pi r for the top and bottom of the cylinder, but I really don’t get this piece. So on this sheet of paper coming out to you now, your job is to explain to absent kid where this 2 pi r h piece of the equation comes from.
CM: Look up real quick. Your homework tonight and the note sheet were just passed out. The note sheet’s a half sheet, grab that, make sure you have it in your notebook for tomorrow. And then the last thing on your homework, the first three are easy, we’ve done these the last two days. Formulas, plug in, solve. Now the last – the one in the back, though? These are the ones I was talking about on your report cards, the type of problems that will make you more intelligent. Do not bail on number four. It’s a thinking problem, don’t just bail on it. You need to put your best effort in order to get credit for the assignment tomorrow. Bye, I will see you tomorrow, great job today!