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Topic: Decimal Multiplication
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Jerry P. Becker

Posts: 13,619
Registered: 12/3/04
Decimal Multiplication
Posted: Feb 4, 2000 2:54 PM
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************************************
From the AMTE listserve, February 3, 2000
************************************
A big part of the difficulty with decimal multiplication is
because we tend to think of multiplication as repeated
addition/equal sets. I am beginning to wonder if a part of
the problem, though, is that, even though we approach
multiplication as repeated addition/equal sets, we no longer
emphasize the difference between multiplier and
multiplicand.

Here is what I observed in Japanese elementary math
textbooks. In Japan, multiplication is introduced as
repeated addition/equal sets. However, they emphasize the
multiplier/multiplicand distinction, to the point if
students write a multiplication sentence with factors in the
wrong order, they would be marked wrong.

But, with this distinction, they seem to be able to
gradually expand children's multiplication knowledge.

They start with the case where multiplicand is a decimal but
multiplier is a whole number. It is not too difficult to
extend the repeated addition/equal sets idea to this
situation. For example, 3 x 1.4 = 1.4+1.4+1.4 [by the way, in
Japan, they write multiplicand first, so they would write
this as 1.4 x 3]. At this stage, they also take advantage of
the metric measurement. So, a typical introductory problem
will go something like:

A juice bottle contais 1.2 liter of juice. How much juice do you have
altogether if you had 3 bottles.

They ask children to think about how they can find the
solutions, and among the methods children come up with is
that 1.2 liter is the same as 12 dl. So, 3 x 12(dl) =
36 (dl), and that's equal to 3.6 liters. So, in the
measurement context, they experience "multiplying."

Then, when they start looking at decimal multipliers, they
start with the case decimal x whole number first. For
example, one textbook series opens this topic with the
problem:

Paper tape costs 180 yen for 1 m. How much will the cost if you buy 3.4 m?

As usual, they ask children to think about different ways to
figure this problem out. They give a few possible methods
in the teachers' manuals: figure out how much 0.1m is, then
multiply that by 34; and, figure out how much 34m would be
then divide the cost by 10.

They then focus on this second method and discuss (review?)
the property of multiplication, a x (nb) = n x (ab) [also
(na) x b = n x (ab)], which probably makes good sense in
this problem context.

Then, they quickly extend to think about decimal x decimal
using this idea.

However, in the example in the book, they have something
like this:
2.17 ---(100 times) --->217 x 2.8 --- (10 times) --->x 28
---- ----
(partial products) (partial products)
--------- --------
6.076 <--- (divide by 1000) --- 6076

Only after this lesson,they introduce the "rule" about
locating the decimal point in the product so that the number
of digits to the right of the decimal point in the product
is equal to the sum of places to the right of decimal point
in the factors.

What I find interesting is that this particular Japanese
textbook does more "teaching" than just giving a rule than
the series Kris' daughter uses. I'm not sure if I agree
with this particular sequencing, and I would have probably
preferred using more of multiplication as rectangular area
idea. But, one thing that strikes me about Japanese
textbook is how well organized it is and how they fit all
together to form a nice cohesive unit. Moreover, even when
they are teaching a procedure, they do seem to try to
establish a sound conceptual base.

Just offering my observations.

Tad Watanabe

Kristin Kaul wrote:

> I have a question for this list.
> Having just read Liping Ma's book, I was amazed to see an example that
> was tailor-made for her book arise in my own home. My 5th grader came
> home with problems in decimal multiplication. At the top of the sheet
> was the explanation: "the number of decimals in the product is the sum
> of the decimals in the factors" - and that was all the teaching they
> received.
>
> My daughter, meanwhile, could see the procedure she was supposed to
> follow, but couldn't really understand why 0.19 x 0.08 wasn't = 1.52
> (i.e. she was multiplying 0.19 x 8 = 1.52). In trying various ways of
> explaining this to her, so many other underlying principles came to
> light that it became difficult to know where to start - clearly they
> had to be made concrete first (place value, multiplication and division
> by 10's, relation between multiplying by a fraction and dividing by a
> whole number..). How do people on this list think this topic should
> have been approached, and what fundamental principles drawn on to
> support it?

***************************************
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
Carbondale, IL 62901-4610 USA
Fax: (618) 453-4244
Phone: (618) 453-4241 (office)
(618) 457-8903 (home)
E-mail: jbecker@siu.edu

mailto://jbecker@siu.edu





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