Title
What Happens When…?
Problem Statement
What happens when you multiply a number by a number less than 1? Consider all cases.
(How do you know you have considered ALL cases?)
In general, what happens when you multiply a whole number by 0.05? In general, what
happens when you multiply a whole number by 0.48?
In general, what happens when you multiply a whole number by 0.9?
Problem Setup
For this problem we shall use the aid of a table to generate data. Following each table shall
be am examination of our results.
Plans to Solve/Investigate the Problem
For “a number”, we shall limit our choice to one positive whole number. We understand this
may somewhat limit any generalizations we might otherwise draws. Ideally we might
choose to include both negative and decimal representations for “a number”. Instead we
shall choose to focus on other parts of the problem.
A “number less than one” will consist of positive and negative 0.75, 0.5, 0.25, 0, and some
integers less than zero.
Investigation/Exploration of the Problem
What happens when you multiply a number by a number less than 1?
Number
Less than
A Number One
Product
4
0.75
3
4
0.5
2
4
0.25
1
4
0
0
4
-0.25
-1
4
-0.5
-2
4
-0.75
-3
4
-1
-4
4
-2
-8
4
-3
-12
4
-4
-16
4
-8
-32
When we multiply “a number” (for “a number”, we chose positive 4) by “a positive number
less than one”, we get a positive product less than “a number” but greater that “a positive
number less than one.”. “A positive number” * a positive number less than one < “a
positive number” but > “a positive number less that one”. The product between “a number”
and “a negative number less than one” is a negative number less than both “a number” and
“a number less than one”. “A positive number” * “a negative number less than one” < both
“a positive number” and “a negative number less than one”. The fact the products are all
whole numbers or integers is arbitrary. There are many other numbers less than one we
could have chosen that would allow for a decimal product. Also, our results are limited by
the fact we only chose a positive number to represent “a number.” The “ “ symbols are
included to represent a specific number described in general terms.
In general, what happens when you multiply a whole number by 0.05?
Multiplication of a Whole Number by 0.05
A Whole
Number
0.00
1.00
2.00
3.00
4.00
8.00
16.00
32.00
96.00
288.00
* 0.05
0
0.05
0.1
0.15
0.2
0.4
0.8
1.6
4.8
14.4
When we multiply a whole number by 0.05, the result is always a smaller positive result. 0
< result from a positive number * 0.05 < the positive number. As the whole numbers
increase by one, the result of the number * 0.05 increases by 0.05. As the whole numbers
double or triple, so do the result of the numbers * 0.05. If you were to multiply each of the
results from the second column by 20 you would get the corresponding whole number from
column a. This is due to .05 written as the fraction 1/20 which is the reciprocal of 20.
In general, what happens when you multiply a whole number by 0.48?
Multiplication of a Whole Number by 0.48
A Whole
Number
* 0.48
0.00
0
1.00
0.48
2.00
0.96
3.00
1.44
4.00
1.92
8.00
3.84
16.00
7.68
32
15.36
96
46.08
288
138.24
When we multiply a whole number by 0.48, the result is always a smaller positive number
less than 0.5 of the original number. As the whole numbers increase consecutively, so do
the results yielded increase by 0.48. When the whole numbers double or triple, so do the
results when multiplied by 0.48. However, for every increase by one of the whole number,
the result yielded when multiplied by 0.48 decreases by 0.02 from 0.5 of the original
number. As the whole numbers increase, the result yielded when multiplied by 0.48
diverges from 0.5 of the original number in a negative direction, or towards zero.
Multiplication of a Whole Number by 0.9
A Whole
Number
* 0.9
0.00
0
1.00
0.9
2.00
1.8
3.00
2.7
4.00
3.6
8.00
7.2
16.00
14.4
32
28.8
96
86.4
288
259.2
When we multiply a whole number by 0.9, the result is always a smaller positive number.
As the whole numbers increase consecutively, so do the results yielded increase by 0.9.
When the whole numbers double or triple, so do the results when multiplied by 0.9.
However, for every increase by one of the whole number, the result yielded when multiplied
by 0.9 decreases by 0.1 from the original number. As the whole numbers increase, the
result yielded when multiplied by 0.9 diverges from the original number in a negative
direction.
Author
Pat Devane
Memorial Middle School
pjdevane@rockdale.k12.ga.us