helping students understand the distributive property
TRANSCRIPT
Helping Students Understand the Distributive PropertyAuthor(s): Steven SchwartzmanSource: The Mathematics Teacher, Vol. 70, No. 7 (OCTOBER 1977), pp. 594-595Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27960973 .
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sharing teaching ideas
Helping Students Understand
the Distributive Property
(a + b)2 = a2 + b2
y/x2 ?
y2 =
x/x2' -
y/y2 = ?
y
a(xy) = ax ?
ay
These are three mistakes that my stu dents frequently make. All three errors in dicate an inability to recognize when one
operation is distributive over another. To offset this problem, I have found it helpful to organize operations into two categories called "increasing" and "decreasing," with each category subdivided into three levels
(table 1). Level I is addition; level II is
multiplication (repeated addition of the same number); and level III is ex
ponentiation (repeated multiplication by the same number).
The hierarchy of decreasing binary oper ations is similar. Level I is subtraction (the most basic decreasing operation); level II is division (repeated subtraction of the same
number); and level III is finding the root of a number (essentially repeated division).
TABLE 1
Level I Level II Level HI Increasing + y
? y xy
Decreasing ?
y -
y tfx
The increasing operations at levels I and II are commutative, whereas all the others are not. Also, operations at level III require more explanation and more practice than the other, more familiar operations at levels
I and II. This additional drill may be needed because of the difficult notation. If, as in the BASIC computer language, 23 is written 2 \ 3, with an explicit operational sign, it might be easier to understand. Like wise \f% can be written 8 J, 3. These sym bols themselves strongly suggest that taking a root is the inverse of raising to a power.
Students should have a great deal of
practice rewriting expressions with explicit operational signs. For example, 2(3)(4) is rewritten 2-3-4, and 53, or 5 \ 3, is rewrit ten 5-5-5. Once students are thoroughly familiar with all six operations (and the
symbols used to represent them) and the levels in table 1, we can describe the distrib utive property in a new way. Distributing is
applying an operation from a given level to two quantities related by operations of the next lowest level only.
In
5?( - y)
= 5- : + 5-y,
a level II operation, multiplication, is ap plied to both the quantities related by a level I operation, addition. In
(xyY = x4y\ a level III operation is applied to a level II
operation. But
\/x + y \/x +\fy or
{x + y)\2ixl2+yll because a level III operation cannot be dis tributed over a level I operation. Distrib
594 Mathematics Teacher
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uting any operation over itself does not result in true statements either. For ex
ample:
x(yz).? (xy) ? (xz)
This method of studying the distributive
property may seem wordy at first, but it does seem to assist students in organizing
their thoughts, and it does help their under
standing of the order of operations. Of
course, they could also test an equation illustrating the distributive property by supplying several numerical examples.
Steven Schwartzman Austin Independent School District
Austin, TX 78751
Converting from Base 10: Nonintegral Bases?
One of my students, Anita Garcia, devel
oped an interesting algorithm in response to the question,
"Find the base b so that 44ten = 62 ,."
Anita's algorithm is as follows: Subtract two from both sides so that the base b numeral ends in 0.
42ten = 60,
In base ten divide the numeral written in base ten by the digit in the tens place in base b.
so
42 6 = 7,
b=l.
Does the algorithm work all the time? Note that
( 2 . ? . Xn)ten ~
(xy)b = xb + y
( 2 ? . Xn)ten ~~
y _ ^
X
But what if the base b numeral had more than two digits? For example,
48ten =
120,.
Using a variation of the rule above, we try
48 -5- 12 = 4,
so
6 = 4.
Unfortunately, the correct answer is b = 6.
Before giving up, however, note that
n2)four = 6.
Is this a new algorithm? Will it work all the time? Let's try.
Now
and
but
72ten = 120,
72 -s- 12 = 6
(12)six =
8,
Well, what is 6?
72ten =
120, 72 =\-b2 + 2-b + 0
b2 + 2b - 72 = 0 b = -1 ?x/13.
What's this? Nonintegral bases? The fail ure of the algorithm raises many questions about polynomials, factoring, and bases that are not whole numbers.
Does Anita's algorithm work on the
problems that do not involve base 10?
37eight = 51,
36eight ^ 5??
36eight -s-
5eight =
6eight and b = 6.
Will it work all the time?
October 1977 595
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