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?

A Number / Number Less than 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 / * 0.05
0.00 / 0
1.00 / 0.05
2.00 / 0.1
3.00 / 0.15
4.00 / 0.2
8.00 / 0.4
16.00 / 0.8
32.00 / 1.6
96.00 / 4.8
288.00 / 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