Avoiding Careless Mistakes
Date: 01/30/2002 at 03:04:15 From: David Subject: How to avoid careless mistakes How to avoid careless mistakes? I have tried do as many problems as possible, but mistakes are constantly made just because of carelessness! I would REALLY appreciate your helping me solve this problem.
Date: 01/30/2002 at 11:10:05 From: Doctor Ian Subject: Re: how to avoid careless mistakes Hi David, I used to have this problem, and in my case the first step toward getting over it was to read _The Gift of Fire_ and _Less Than Words Can Say_ by Richard Mitchell. You can find them online here: The UnderGrammarian - Richard Mitchell http://www.sourcetext.com/grammarian/ If you'd like to own them - and I highly recommend this, since it's a good idea to read them once a year - you can order them from www.commonreader.com. Self-help guru Anthony Robbins teaches that the hard part about changing your behavior isn't learning the new behavior, but getting a very clear understanding of why you really _need_ to change the old behavior! That kind of understanding is what I got from reading Mitchell's books. Once I became convinced of the importance of what he calls 'the habit of correctness and precision', I found that I started adopting it quite naturally, without much effort at all. Having said that, it seems to me that many, if not most, of the careless mistakes that we see here at Ask Dr. Math are caused by trying to do too many steps at once. For example, 3(x - 5) = 18 3x - 5 = 18 3x = 23 x = 23/3 Wrong! Here, the multiplication wasn't fully distributed over the addition in the first step. This is an easy mistake to make, and the way I avoid making it is to insert an extra step, in which I leave the parentheses in place: 3(x - 5) = 18 (3*x - 3*5) = 18 3x - 15 = 18 3x = 33 x = 11 Check: 3(11 - 5) = 3(6) = 18 These 'extra steps' can be especially useful when dealing with negative numbers, e.g., 3(x - 5) - 4(x - 6) = (3*x - 3*5) - (4*x - 4*6) = (3x - 15) - (4x - 24) = (3x - 15) + -1(4x - 24) = (3x - 15) + (-1*4x - -1*24) = (3x - 15) + (-4x - -24) = (3x - 15) + (-4x + 24) = 3x - 15 + -4x + 24 = Now I can be pretty sure that I have the signs correct. This is more work than something like 3(x - 5) - 4(x - 6) = (3x - 15) - (4x - 24) = 3x - 15 - 4x - 24 = which is much faster, but which also happens to be wrong. A good rule to keep in mind that you can't make mistakes fast enough to get a correct answer. :^D I also make it a point _always_ to check my answer, after I think I've found it. At first this is something you have to remember to do, but after a while it becomes natural. A second kind of careless error, which sort of falls into the same category, is caused by translating story problems too quickly into equations. For example, here is the kind of thing we see a lot: The sum of half of a number and 8 less than the number is 25. (1/2)N - 8 = 25 (1/2)N = 33 N = 66 Wrong! The mistake can be found with a careful step-by-step translation from English: The sum of half of a number and 8 less than the number is 25. \______/ \________/ The sum of half of N and 8 less than N is 25. \___________/ The sum of (N/2) and 8 less than N is 25. \________________/ The sum of (N/2) and (N - 8) is 25. \__________________________________________/ (N/2) + (N - 8) is 25 \/ (N/2) + (N - 8) = 25 which leads to (N/2) + (N - 8) = 25 (N/2) + N - 8 = 25 (N/2) + N = 33 (3/2)N = 33 N = (2/3) * 33 N = 22 Check: The sum of half of 22 and 8 less than 22 is 25. The sum of 11 and (22 - 8) is 25. The sum of 11 and 14 is 25. Yep! At each step, I identified one small piece of the sentence to be translated into mathematical symbols. I made the translation, and then continued as before. If you were doing this on paper, you might do it more quickly by crossing things out and writing their replacements above or below, e.g., The sum of half of xxxxxxxx and 8 less than xxxxxxxxxx is 25. N N The sum of xxxxxxxxxxxxxxxx and xxxxxxxxxxxxxxxxxxxxxx is 25. (1/2)N N - 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx is 25. (1/2)N + N - 8 and so on. (Here, 'xxxxx' would really look like the original words with a line drawn through them to show that they no longer need to be considered.) On the one hand, working 'in place' is an easy way to get sucked into making careless mistakes; but on the other hand, the more times you copy something, the greater the error that you're going to copy it incorrectly. With a computer, you can simply copy the old line and change it, which is what I've done here. Without a computer, you have to use your judgment about whether working in place or copying is more likely to cause a problem. To make that judgment, you have to have some idea about the kinds of mistakes that you tend to make, and how often you make them. Which leads to my final recommendation, which is that you might want to keep a notebook of the careless mistakes you make. Keeping track of them would allow you to observe patterns, and figuring out what you're doing is the first step towards changing _any_ kind of behavior. If you find that there are certain _kinds_ of mistakes that you're making over and over, feel free to write back and tell us what they are. That would help us come up with techniques that you could use to avoid making them in the future. I hope this helps. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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