Integer Function
Problem Statement
Find the smallest
integer x for which
represents an integer.
(Source: Mathematics Teaching in the Middle School, SepOct 1996).
Problem setup
This problem is asking me to
find a number (the smallest possible) – here represented by x that
will make the fraction equal to an integer.
Plans to
Solve/Investigate the Problem
Initially, I thought about the
number 0; however, this is a sophomoric mistake of great
proportions!!! (Remember, I am
full of drama!!)
Investigation/Exploration
of the Problem
While using x = 0 will give me
an integer result, 0 is not the SMALLEST value we can use. Consider negative values. Some of these will also give integer
values. Here is a chart for
your review:
Value

Result

1

6

0

12

1

#DIV/0!

2

12

3

6

4

4

5

3

6

2.4

7

2

8

1.71429

9

1.5

10

1.33333

11

1.2

12

1.09091

13

1

14

0.92308

15

0.85714

16

0.8

17

0.75

18

0.70588

19

0.66667

20

0.63158

21

0.6

Notice that for several of the
negative values, the result is indeed an integer! When x = 13, the result is 1. This is indeed the smallest value of
x that will yield an integer result.
This seems rather simple to me,
so I would like to pose an additional question: What would cause the result
to be zero? Let’s explore the world of calculus!
For an interactive lesson on
this, click
here. Be sure to turn your
sound on!
Could we take the limit of this
expression as x approaches infinity?
Ponder this… as the denominator grows more
negative, the decimal result gets smaller! As x approaches infinity (that
sideways 8 over there represents infinity), the quotient gets closer and
closer to 0.
Extensions of the Problem
Discuss possible extensions for
the problem and explore/investigate at least one of the extensions you
discussed.
Author & Contact
mhouston@rockdale.k12.ga.us
