Estimation is a bit of a lost art, or it can be among students. Projects like Estimation 180 are hoping to change that. And you might wonder, ok that site is cool and everything, but I've got state testing to worry about I don't have the time to spend figuring out the cost of a dollar sign some blogger weirdo made out of pennies. And yet, if I'm a struggling student, being able to make some estimates might save my chances of graduating.
[Problems taken from Texas Education Agency study guides]
Any geometry teacher will instantly recognize that this is a sector area problem. The answer can be determined exactly if you know how to relate the total area of the circle to the percentage of that total represented by that 170º chunk. The work would look something like this (steps exaggerated to mirror the steps a student would go through).
Assuming the student remembers the proportional relationship between a chunk of circle and the entire circle, they'd probably work the problem similarly. There's room for some algebra mistakes. But let's say you're not the best math student, you have no idea what formula you should use and your formula chart will you how to find the area of a complete circle, but doesn't offer any hints about chunks of circle. Well, if they've been taught how to make initial guesses via estimation, this might be the thought process:
I see the word area in the problem and run with it. Formula chart tells me what I need for the area of a circle. That shaded section is like maybe half the circle, a little less. There's only one answer choice that is less than half of my number.
The next problem drives this home even more.
For a lower performing student, this one can be a bit intimidating. The word "midpoint" helps and the formula chart does have a formula for this. Using it will yield the exact answer.
Not too bad. But again, imagine being the C-student taking this test. Look at the algebra hurdles. Look at the connection that has to be made between those large terms in the midpoint formula and what's been given. As long as we have a conceptual understanding of midpoint, a drawing is a much faster way to approach this.
I'm given some coordinates. I know how to plot coordinates. If M is the middle, B should be like a similar distance away. The answers are coordinates. I know how to plot coordinates. Only one of these coordinates seems to match the picture I drew.
You might call this subversion of the standards that this test is trying to measure. I ignored the formula for sector area. I ignored the midpoint and distance formulas. And yet estimation got me to the same place. If you flip through any standardized math exam given in Texas in the last 10 years, a LOT of the math problems can be solved with first order estimation.
Does that mean we stop teaching the standards? Nope. Does that mean we should change our opening acts when it comes to the standards? Probably. Both of my estimation methods could easily be openers to the concept of sector area and midpoints. But a textbook problem would prefer if you jumped immediately to algebra.