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16 Sep 2005 - 12:36
What is subtraction? Subtraction is the ______________ of addition.
When you subtract, you __________ ___________ ___________________ of the number you are subtracting.
An absolute value is always _________________.
1 - 2 = _________ 1 - ( - 2 ) = _________ -1 - 2 = _________ -1 + -2 = _________ 1 - | 2 | = _________ -1 - | 2 | = _________ -1 - | -2 | = _________
study sheet for class quiz on pages 2 - 16, Prentice Hall Mathematics: Explorations & Applications & Prentice Hall Pre -Algebra outloud sheets: integers & absolute value
notes on outloud sheets for integers & absolute values
Carolyn on introducing absolute value
keywords: integers subtraction addition absolute value opposite add study sheet outloud out loud
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I wrote a little about integers here: http://www.mathandtext.blogspot.com/2005/08/operating-with-negative-numbers.html although I focused on multiplying and dividing. I have been reading your posts about negative numbers and have been having random thoughts all over the place that I haven't quite put together yet. I guess my first reaction was surprise that Christopher's class didn't start off with the seemingly very popular "zero pairs" approach to teaching addition and subtraction of integers. This method uses two colors of chips--red and yellow usually. Each red chip is -1, and each yellow chip is +1. To find, say, 5 - (-7), texts show starting with 5 yellow chips (positive chips): [+] [+] [+] [+] [+] The task is to subtract 7 red chips (negative chips), but since we don't have 7 red chips, one puts in zero pairs until one has 7 red chips to subtract. A zero pair consists of one yellow chip and one red chip. We can include these into any problem we're working on, because the value of the group is 0 [+1 + (-1)]. Thus, if we add 7 zero pairs to our five positive chips above, we do not change the value: [+] [+] [+] [+] [+] [+] [+] [+] [+] [+] [+] [+] [-] [-] [-] [-] [-] [-] [-] Now we have the requisite 7 negative chips to subtract. And, of course, after we do that, we see that our difference is +12: [+] [+] [+] [+] [+] [+] [+] [+] [+] [+] [+] [+] This model is interesting, but I think it is yet another example of a false implication. First of all, there is no such thing as a negative quantity, as the model above suggests. In my mind, one can properly conceive of integers as "directional values." (Negative signs can also be thought of as indicating opposites.) They do not represent quantities per se. The integer -2 means "2 to the left of zero." The integer +2 means "2 to the right of zero." The expression -(-2) can mean "the opposite of 2 to the left of zero." Addition signs and subtraction signs take on directional meanings too, I think. When we add and subtract integers, we don't have to start at zero. So 5 - 7 means "7 to the left of 5." The expression -5 - (-7) would mean "the opposite of 7 to the left of -5," which of course means "7 to the right of -5." In a nutshell, I think the number line model properly communicates an accurate understanding of adding and subtracting with integers. By the way, if we understand integers as directional values, then the response desired on the study sheet: "An absolute value is always [positive]" would be incorrect, really. -- JdFisher - 16 Sep 2005
J. D., My daughter encountered the two-color chip thing in Scott-Foresman's 6th grade book last year. It was the first math thing she's done that has thrown me completely off balance. I tried to skim the text to get it, but just couldn't make sense of it. Months later, I don't know if she would be able to answer those types of questions. I know I would still have a lot of question marks floating over my head. -- DanK - 17 Sep 2005
By the way, if we understand integers as directional values, then the response desired on the study sheet: "An absolute value is always [positive]" would be incorrect, really. I've found it useful to think of the integers as real numbers (i.e. numbers with positions on the real line, and hence a direction) and absolute values of numbers as magnitudes, i.e. indicators of size. In reality there is no difference, and the absolute value is actually a non-negative real number. -- CarolynJohnston - 18 Sep 2005
I want to take this test! At least the verbal parts. It reminds me of mad libs. Subtraction is the (noun) "ugly stepdaughter" of addition. When you subtract, you (verb) "stink" (definite article) "the" (noun) "cheese" (preposition) "up" of the number you are subtracting. An absolute value is always (adjective) "puny". How did I do? -- CarolynJohnston - 18 Sep 2005