Decimal Multiplication and Division
Date: 11/19/2002 at 01:34:04 From: Susan Subject: Decimal division, with zero in tenths place Hi Dr. Math, I'm tutoring a 6th-grade student, and I'm having a hard time explaining to her why you sometimes need to put a zero in the tenths place, in the quotient - as a sort of placeholder, I tell her. For example, we were dividing 0.5 into 0.025. She's okay with multiplying both numbers by 10, so that the problem becomes 0.25/5, but when we start writing digits for the quotient, she won't believe that the answer is 0.05. She insists that it must be 0.5. I've tried showing her that 0.5 x 5 = 2.5, and not 0.25. And I've tried using 25 base 10 cubes to represent 25 one-hundredths, and then having her divide them into five groups, each of five one-hundredths. She still says you cannot put that zero in after the decimal point. It seems like something is just missing. For one thing, she has it firmly in mind that her teacher said to never add a zero after the decimal point, so she refuses to believe any of my explanations. (For another, she's still a bit shaky on decimal multiplication.) Do you have any other way of explaining decimal division when the numbers that are not strictly analogous to regular division - i.e., when you have to put in a zero in the tenths place, or hundredths place, etc.? (I suspect my student is not the only one who could use help in this area.) I would appreciate any help you could provide. Thank you, Susan Tutor in San Francisco Bay Area
Date: 11/19/2002 at 13:04:09 From: Doctor Peterson Subject: Re: Decimal division, with zero in tenths place Hi, Susan. It sounds as if the student is stuck on rules and doesn't dare try to understand what's going on. The claim that "you cannot put that zero in after the decimal point" must come from something the teacher said in some other context, and she feels math is all about rules. That probably happens more often than we'd like to think; too many teachers emphasize rules rather than reasons. So let's first see where that rule might come from, and maybe we can untangle it from this problem. All I can think of is just that once you have a number, you can't insert a zero between the decimal point and other digits without changing its value. You'll have to ask her to give an example where her teacher might have used that rule, and then show why it doesn't apply here. I explain decimal multiplication and division in terms of fractions (assuming a student has an understanding of fractions!). In this case, 25 5 25 10 25 * 10 10 25 0.025 / 0.5 = ---- / -- = ---- * -- = -------- = ---- * -- 1000 10 1000 5 1000 * 5 1000 5 So the answer will be 1/100 of 5, or 0.05. This will probably be too complicated for your student. Here's an alternative way to show the same thing: 0.025 1000 * 0.025 25 25 1 ----- = ------------ = --- = -- * --- 0.5 1000 * 0.5 500 5 100 That is, we can simplify the fraction (that is, get the same answer to a simpler division) by multiplying both numerator and denominator (dividend and divisor) by 1000; and then we can take a factor of 100 out and leave it for the end, when we will place the decimal point. The root of this is to separate out the actual long division from the placement of the decimal point. The usual way, which you have been using, is to just get rid of the decimal point in the divisor, and leave one in the dividend, knowing that you just place it in the quotient directly above. But since your student has sort of a phobia about zeros, doing the division and then saying you have to move the decimal point two places to the left may help. But this may simply isolate the actual problem more clearly. I am geussing she will object just as strenuously to adding a zero in the last step: 5 * 1/100 = 0.05 It might be a good idea to focus on this before you get back into division. This is where you MUST add zeros, and until she sees that, she will not accept anything you say! She really needs a solid understanding of what decimals MEAN, and of placement of the decimal point by multiplying or dividing by powers of ten, before she will be able to really understand multiplication or division. You may find some good ideas in this area in our archives: Elementary Fractions and Decimals http://mathforum.org/library/drmath/sets/elem_fractions.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 11/22/2002 at 17:51:03 From: Susan Subject: Thank you (Decimal division, with zero in tenths place) Thanks Doctor Peterson - - both for your ideas and your quick response. These are great suggestions, and I'll try to use them the next time I see the student. - Susan
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