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Write-up


Title
Ordering Fractions

Problem Statement
Ken claims that 2/3 is between 1/2 and 3/4 since 1 < 2 < 3 and 2 < 3 < 4. He says this always works. Does it?

Problem setup

I will need to substitute the sequential numbers in the numerators and the denominators to see if the pattern still exists.  The denominator of each fraction will always be one more than its numerator.

 

Plans to Solve/Investigate the Problem

I am going to start by substituting small numbers:  4/5, 5/6, and 6/7.  Check to see if the result is the same.  Next I will substitute larger numbers, 111/112, 112/113, and 113/114, and check the result.

 

Investigation/Exploration of the Problem

4/5 is less than 5/6 which is less than 6/7.  I know this because as the numbers increase, the size of the missing piece from each fraction decreases.  That leaves a larger fraction than the one before it.  111/112 is less than 112/113 is less than 113/114.  The reasoning is the same.  I can picture in my mind's eye, the size of the missing pieces.

 

Extensions of the Problem

Next I tried negative fractions, -1/2, -2/3, and -3/4.  I drew a number line and plotted the negative fractions on it to have a picture for us visual learners.  I found the results to be the opposite:  -1/2 is greater than -2/3 which is greater than -3/4.  So Ken's hypothesis will work for positive fractions but not for negative fractions.