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