Estimation Before Computation
You don't need to be a baseball fan to know that the distance from third base to first base is a lot less than three miles. Yet that's what several students came up with (16,200 feet to be exact) in a class I visited a few months ago. And though it may seem like their answer came out of left field, these students were actually just a calculator button away from hitting a home run on a pretty challenging problem. In other words, they knew they were dealing with a right triangle whose legs were both 90 feet long (the distance between home plate and first base, and home plate and third base), and they applied Pythagorean Theorem accordingly. They just stopped a step short (hmmm... a short stop?) by failing to take the square root of 16,200, and were thus off by some 16,073 feet.
What bothered me most about this wasn't that students skipped a step, but that they accepted 16,200 feet as the correct answer--until I said, "So what you're telling me is that it's 90 feet from third base to home plate, and 16,200 feet from third base to first base. No wonder some of those guys take steroids." To which a couple of kids replied, "That can't be right." And after a minute or two of troubleshooting--with no help from me--students correctly changed their answer to 127 feet.
This experience illustrates why it's important for students to treat most if not all math problems as estimation problems. Model this before you work through a problem with students by asking, "Approximately what would you expect the answer to be?," and then recording their estimates. Do this until students start doing it themselves as they solve problems on their own. And if they're resistant or forgetful, you might want to make it a required first step in the problem-solving process.
Whatever it takes to get students in the habit of estimating before computing so that they have a reliable ballpark answer to compare their actual answer to for reasonableness. And so that they can then recognize when they've committed an error before it's too late to fix it.
Image provided by Phillip Martin with permission
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