I was hanging out with some 3rd Graders today. Their task was to find the area and perimeter of some shapes. This girl was working with a shape like this one, except not quite this one:

array1

In particular, the shape she was working with had a perimeter of 22, and an area of 21. She, however, had counted an area of 21 for both. She had already called me over once to ask whether you count each of the sides of a corner square in the perimeter. My Spider Sense went flaring, but I wasn’t sure how to help, so I told her that each side did count for the perimeter. She seemed suspicious, but went with it. Then she called me over again.

Girl: Wait…how is it possible for the perimeter and the area to be the same?

Me: That’s a really interesting question. I’m curious: what’s perimeter, anyway?

Girl: Well… Perimeter…It’s a thing, but it’s outside of something.

Me: Hehe. That’s not a bad start…

Girl: OK, so perimeter is the space outside of a shape. Area is the space inside a shape.

Me: Cool, that’s very interesting!

 

At this point I’ve sort of constructed a theory about her initial question. Maybe she’s thinking of perimeter as 3D space, instead of lined space. That could explain her confusion about the space around the shape being equal to the space in the shape. Maybe that’s also the source of her doubt about double counting the sides of a square in the perimeter. (Though that doesn’t fit in super-well.)

I decided to push on her definition.

Me: So, perimeter is the space outside of a shape. So is this all the perimeter? [I drew a shape and shaded in the area around it.]

array 3

 

Girl: No! No, it couldn’t be, because then that would go on for all of this space. It’s more like this:

array2

Me: Interesting! I have a question about your picture. Is it important that the lines stick out of the shape? Could you have drawn it where the lines don’t extend out of it?

Girl: No, it’s important that they stick out.

At this point I didn’t know exactly what to do, so I just tried to explain that perimeter and area measure different things. I gave concrete examples of perimeter (“It’s like a fence”), but I didn’t really feel like the explanations stuck with her.

I’m curious to hear all of your thoughts on this interaction. My takeaway is a curricular one. Area and perimeter are different concepts, and they don’t necessarily benefit from being presented together and in contrast with each other. Maybe it’s better to introduce each individually, and only play them off each other after students have a solid notion of each concept’s meaning.

  • kmorrow-leong

    Maybe if we imagine it done with square tiles. The tiles inside are the area. The tiles immediately outside of the figure are the perimeter. That would be represented by (L+1) + (W+1) + 4. It’s hard to tell without the actual figures and a demonstration!
    Interesting.

  • Michael Paul Goldenberg

    Makes me wonder a bit if she’s thinking of perimeter as a border that is also 2-D, like a walkway around a pool, rather than just the distance around the pool. Can’t say her extended line drawing rings any bells from my experience with other students, though.

    I tend to think that this is one of those really key ideas we don’t generally do a good job with in school mathematics in the US: the notion of dimensions and that certain sorts of measures and certain dimensions go together. So when I get older kids struggling with this, particularly as they’re loaded down with all these different geometry terms that they don’t really have a handle on by the time I’m working with them on ACT/SAT prep, or tutoring for algebra or whatever, I try to get them to categorize into 1-D, 2-D, and 3-D the relevant terms: perimeter, edge, circumference, radius, diameter, etc., all go together. Area and surface area go together. Volume is alone in 3-D for their purposes.

    I try to have them think about whether the particular thing they’re discussing can be stretched into a straight line, sliced off what it’s on and flattened (if it’s not already flat), or neither. It’s not foolproof by any means, but it seems to help some kids get a better visual sense of what’s what.

    Of course, the very derivation of ‘perimeter’ comes from two Greek words/prefixes that mean “measure around” which could be helpful for some kids.

    But for 3rd grade, I think it has to come down to seeing perimeter as the length of what you’d get if you took a string that traced the exact path around the outside of something. and then stretched it into a perfect straight line. Those other words in the 1-D category are just variations (circumference) or straight lines already.

  • The fence idea is good but still a bit abstract. I suggest putting her paper on a cork board put 4 pushpins at each corner, handed her a string, and had her rope her square to experience the perimeter herself. Then ask her to untie her “fence”, measure it, et voila! While the shape is roped hand her a big glob of playdoh, let her knead it to fit her roped square, the have her decorate the playdoh with 1-cm square tiles grooved with a straight edge, and have count squares. My two cents worth of ideas. Thank you for the cool work you do with your students!

  • I think Michael Paul Goldberg above is getting at the real problem and , in my mind, it comes down to units. Even at the AP level, it seems like it is a coin toss for my kids as to whether the area of a circle is 2 *pi * r or whether it is pi * r^2 I try to get them to tell me what units area is measured in and see that the unit forces their hand in choice. My son is in 4th grade and I am pretty confident that the unit-based approach would completely elude him.

  • One thing I’ve been wondering about is, in addition to what Michael Pershan wonders about introducing area and perimeter separately, whether we move away from the real meaning of “unit” too quickly.

    Taoufik mentions the cool idea of students using string to trace the perimeter and measure that… but there are a few layers of abstraction happening there between the question of “how many linear units of measure does it take to measure all the way around the outside?” and “how long is the string that goes all the way around the shape?”

    I think that before students measure with rulers and protractors and yardsticks and tape measures they should measure with little baggies of units. Just like they measure area with unit squares and volume with unit cubes, they should have little baggies of linear units (inches, centimeters, feet) that they use to measure. They should count out actual units, lining them up next to linear things. The same thing with angles: they should have little baggies of one-degree angles that they line up inside of the angles they want to measure. Only when they complain about the cumbersome-ness of individual angles should they be invited to use tools that have the measurements already glued together. Because I don’t think students connect measuring with counting units, and so the whole idea of units becomes meaningless.

    • My vote: introduce the concepts of perimeter and area separately at such an early grade level. I like Max’s idea of units inside baggies, but I’ll throw an extension to Max’s idea. If I’ve learned anything from File Cabinet, a sticky note can be a unit of measurement without even considering customary units of measurements like inches and centimeters. What if students used linear objects such as toothpicks, straws, or spaghetti as units to measure items in a classroom?
      Hey guys, let’s find how many toothpicks it will take to go all the way around this giant book I found at the library? Or how many straws will it take to go all the way around the walls of our room? I believe these are cheap supplies that will get you some mileage with student involvement and conceptual understanding of perimeter.

  • Beth S

    From her illustration, I think she has somewhat of a grasp of the concept of perimeter. It seems that she wants to use the same unit of measure that she used to measure the area of the rectangle so she extended the grid lines to show to you the iteration of the unit around the shape.

  • Something else to consider would be to approach this problem with polygons that have inside corners, something like a U shape. If you made a small one and had students discuss their different answers (where some students single count the inside corners as it is just one square), I think some great conversation would come out of it.

  • Not enough measuring going on. If one is fixated on squares, draw a curved shape on squared paper and ask the question.

  • Jasper

    Her answer is based on counting the area of each unit square outside the figure that shares a side with the figure. In other words, for each segment of the figure, she makes a rectangle (one unit high) whose base is the segment. She then adds up the areas of the rectangles to get the (one unit) times (the true perimeter).

    She needs to learn that the units of perimeter are units of length, whereas the units of area are length^2. This will require her to divide her (unit times perimeter) by the unit, to get the true perimeter.

    Or she can learn to count the length along the outside, without needing to create rectangles and divide by the unit length.

  • Deez Nuts

    Deez nuts