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?
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
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
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
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.