Perimeter and Area: The Bane Of My Existence

Recalling back to my days of youth, I can remember some challenging issues involving my "understanding" of math. There was the whole algebra thing. A+B=C, right? So X+Y=Z, correct? Variables... ugh. It's embarrassing to think of me having such a hard time grasping those concepts. If someone had just told me, "It's like having missing numbers, and you try to find the number that fits," I might have been spared months of anguish. Instead, I felt like I was trying to decipher some new alphabet. When can A also be a C, anyway?

Don't even let me get started on the whole geometry thing. Proofs. Corollaries. What kind of math was that anyway? I had never done so much writing in math before. Of course, it would've helped if I hadn't goofed around so much.

Maybe this is my penance.

You see, perimeter and area were such easy concepts for me. I'd even venture to say I learned them both in a day. And when I say I learned them both, I mean that I understood the concepts behind them. I like for my students to have a grasp on the "concept" of what I'm teaching. I could easily say, "for perimeter, add all the sides. For area of a rectangle or square multiply the length by the width." But no, I like for them to explore the concepts in more detail. And the more I let them explore, and the more I try to guide them to this "elusive" concept, the more convoluted it becomes. I've tried teaching the two concepts separately. I've tried teaching them back to back. I've tried teaching them jointly. But, for all intents and purposes, I might as well be teaching the theory of relativity.

I use blue painter's tape to tape off outlines of various shapes on my classroom floor (which has tiles that are 1 foot x 1 foot). I have them explore perimeter and area that way. I have them explore perimeter and area on 1 cm x 1 cm grid paper. Most of the times, however, I feel so frustrated, I'd like to put my students on the other side of the perimeter of my classroom.

So, I finally start to feel like I'm making some headway with them. I decide to give them a more rigorous and challenging activity. I give them centimeter grid paper. On the board I write 4 or 5 problems that ask them to make rectangular shapes on their grid paper that meet certain criteria. For example: "Draw a rectangle with an area of 20 square centimeters and a perimeter of 18 centimeters." The stuff I get back blows my mind. Either they ignore the first part of my criteria or the last part. I'll get a rectangular shape that is 2 cm x 10 cm, which meets the criteria for area. But, it does not meet the criteria for perimeter. Or, I'll get a rectangular shape that has a perimeter of 18 centimeters, but does not have the correct area. And it's not like I haven't taught them strategies for generating multiple rectangles that all look different, but have the same areas.

Don't even get me started on the fact that, involving perimeter problems, some students just add 2 sides of a rectangle because those are the only sides that are labled. It's an exercise in futility.

All I know is, God help me if I ever need to hire one of them to install carpet later in life. I have a feeling that they won't bring enough rug with them.