I've tried various things throughout the years, and things "stick" for the unit, but later, say the following year or years, I ask the same question, the student reverts back to JUST dividing by 12. Must be hardwired into their heads.
Anyway, this came up again yesterday in geometry class with the following problem:
You want to paint the exterior of a cylindrical container with a 4 inch radius and 15 inch height. Paint costs 86 cents per square foot, how much would it cost.
I had an answer bank on the sheet, and LO AND BEHOLD, their answer was not on there. Hmmmmmm. Then I prompted: be careful with your units. OH! Okay, convert convert. OH! the answer is STILL not on there. Hmmmmmm. The dreaded JUST dividing by 12 dilemma. Anyway, I held up a piece of white paper and basically did what you see here below.
It SEEMED to make sense to the students. I liked the visual and the methodical dividing the side by 12 AND the algebraic equation by 12 right afterward, so they see what happens. It SEEMED to stick, but I'm not going to fall for that again. I'll quiz them again next year or two to see. My optimistic self thinks, "YES! I've solved the problem of world peace." Don't burst my bubble. Anyway, one more example to add to the arsenal.
A funny (and relevant) thing happened a month ago. I was working with a girl (who is home-schooled) who is learning math out of a 7th grade Singapore text, and we got to the section on converting square units (square meters to square cm, etc), and I thought--we should do some similar problems with inches and feet. So I went to my department's library of middle school math texts (3 text book series), and looked for a section that included converting square units. Guess what? That section wasn't there! So from my short survey, most US middle school texts don't teach this at all (whereas they've been converting linear units since second grade). I'm thinking this could be part of the problem.
ReplyDeleteLsquared:
ReplyDeleteThat's just crazy! Hopefully, the teachers are supplementing. Unfortunately, it doesn't seem to stick much.
I prefer to start with a square foot. "How many inches is this side? "12in". How many inches on this side?" 12in" So how many square inches in a square foot? "144sqin."
ReplyDeleteThen switch to metric and 1g H2O = 1ml, 1ml = 1 cm3; 1000cm3 = 1dm3; 1000L = 1m3;
__ cm3 = 1m3;
__ cm3 = 1km3
How many Liters is that 383ci. hotrod engine?
For what it's worth, I don't think anyone teaches square units or cubic units in middle school. Too busy pretending they're teaching algebra -- raising standards, don't cha know.
nice posting . . keep blogging!!
ReplyDeleteI hang out all day long with problem-solving geeks, most of whom are really visual thinkers. Also, I write problem-solving support for middle-school and elementary-school students. Again, we tend to go visual or hands-on with those groups. If you had posed this scenario to me 5 years ago, I would have been thinking algebraically like you did in that nicely-illustrated document. My newly found visual problem-solver got stuck on the following question: does it matter how those 100 square inches are arranged? Can you always fit 25/36 of a square foot into 100 square inches? What if it's a 1-inch by 144-inch square foot? Is that still considered a *square* foot? Is it easier to see in a squarer square foot? And what are the shapes in the original context anyway?
ReplyDeleteFunny what different questions and answers come up when you approach with a different problem-solving strategy!
Max,
ReplyDeleteThose are great questions. They would be nice extensions. ... I'm guessing you know the answers and are just posing the fun thinking questions. I'll have to enhance the next go-round with my geometry kids.
Why did you have to put that sneaky " what about volume" question at the end?
ReplyDeleteWell, I wanted them to do more than just blindly plug into formulas. Also, I wanted to show a realistic situation of how this can be used. For example, if company is mass producing this objects, they need the cost information for budgets.
ReplyDelete