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 recently visited. And though it may seem like their answer came way 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—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, that is, I said to them, “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 students replied, “That can’t be right.” A minute or two of troubleshooting later—with no help from me—students had correctly changed their answer to 127 feet.
This experience illustrates why we should teach students to treat every problem as an estimation problem. Model this before you work through a problem with students by asking, “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 correct it.
(With standardized tests coming up, this is a particularly good time to stress this idea to students. You can be sure, for example, that 16,200 feet would have been a “trap” choice had the baseball field problem appeared on a multiple choice test.)