Intermath | Workshop Support

 Write-up

Integer Function
Problem Statement
Find the smallest integer x for which

represents an integer.

(Source: Mathematics Teaching in the Middle School, Sep-Oct 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!

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