Explaining Subtraction of Integers as Adding the Opposite
Date: 02/16/2007 at 19:41:41 From: Kristin Subject: Real-life applications of integer subtraction Hello, I teach grade 7 and we are currently working on how to subtract integers. I like to give students as many examples of real-life applications as possible, but I haven't been able to figure out how to explain WHY, when subtracting integers, we add the opposite. Is there an explanation or word problem that could help to understand this? We have been relating integers to money and temperature, but I cannot figure out how to do this with a question such as (-5) - (-3). Students want the answer to that problem to be -8, not -2. I have tried to use the "walking the number line" idea with them, but it still doesn't answer their question of "When would we encounter this in real life?"
Date: 02/17/2007 at 13:19:58 From: Doctor Peterson Subject: Re: Real-life applications of integer subtraction Hi, Kristin. Most applications of this are probably hidden in equations; for example, in solving an equation (as part of solving some real-life problem) you might subtract x; you never have to worry about whether x happens to be positive or negative because the same rules apply regardless of sign. This would not be true if we used some different way of subtracting negative numbers! That is, the rules are as they are primarily so that the rules all work and the entire number system is consistent. I'll come back to this abstract aspect of the question in a moment. There are a couple more visible applications I can think of that might make this clearer to start with. The first is using subtraction to find a difference. How much did the temperature change if it went from -3 to -5 degrees C? You find the answer to this question by subtracting; if the temperatures had been, say, 73 and 67, we would subtract the old from the new and get 67 - 73 = -6, meaning the temperature went down 6 degrees. The same happens in our problem: we subtract the old, -3, from the new, -5, to find that the temperature went down 2 degrees: -5 - (-3) = -2. Look at it on a number line, and it's clear the difference should be 2, not 8. Why is the answer -2? The subtraction -5 - (-3) asks what number we have to add to -3 to get -5; and (-3) + (-2) = -5. We can find this -2 by undoing the addition of -3: -5 plus the opposite of -3 is -5 + 3 = -2. This illustrates the abstract meaning of subtraction: subtracting any number means undoing an addition (which is the same as adding its opposite). That is, a - b = c means a = b + c and we can find c by adding -b to both sides of the latter equation: a + -b = b + c + -b = b + -b + c = 0 + c = c So a - b and a + -b are the same thing. This is the actual proof of this fact on an abstract level, as opposed to plausibility arguments based on illustrations. But if you go back to the origins of negative numbers, they were based not so much on abstract considerations as on money. Positive numbers represented assets or gains, while negative numbers represented debts or losses. And taking away a debt (reversing a loss, that is, subtracting a negative number) is equivalent to a gain (adding a positive number). For example, if I owe 5 dollars (net), my assets are -5. Suppose that, among my accounts, there is a debt of 3 dollars, and a friend forgives the debt, in effect taking away that -3 by paying me 3 dollars. In subtracting -3 dollars from my account, he is adding +3 dollars. Subtracting the negative means the same as adding a positive. You can make this idea visible by using play money and IOU slips. If I have IOUs for $2 and $3, making a total of -$5 in my pile, then if someone gives me $3, it will cancel (pay off) the -$3 and leave me with -$2 in my possession. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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