One great way to get students to see that they can list several names for one fraction is by exploring what we could name each section of a clock face that has had the hands placed on different numbers. We always leave one hand at 12, and our clocks have equal length arms so we never know know whether we are looking at an hour hand or a minute hand.

Starting with the bottom clock, we see that one hand is on the 12 and one hand is on the 2. Most students first see this as 2 out of 12 hours, and they write the fraction 2/12. Many also see that this could be 10 out of 60 minutes, or 10/60. From that point, many notice that neither of these fractions is stated in its lowest terms. Most then "chop-chop" 2/12 (divide by 2/2) to get the fraction 1/6. Many then can say that 16 2/3% of the clock face has been "covered up" or "rotated through".

The next picture up shows one hand on the 12 and one on the 8. This shows 8/12 or 40/60. Again, neither fraction is in lowest terms. Many students will just see this as 2/3, as 4/12 looks like 1/3, but to prove it, one could divide 40/60 by 10/10 to get 4/6. Then one could divide 4/6 by 2/2 to get 2/3. We have mathematical PROOF! Most students now see that 8/12 is the same as 2/3, and that is the same as 66 2/3%. SWEET!

The goal in all of this is to be able to add fractions with unlike denominators. The third picture up illustrates the addition of 1/4 + 2/3. 1/4 is seen to be equivalent to 3/12 and 2/3 was proven to be equivalent to 8/12. So, we get 3/12 + 8/12 = 11/12.

We might also get 25% + 66 2/3% = 91 2/3% but 91 2/3 / 100 is not in its lowest terms, and most middle school teachers would simply freak-out if the answer is presented in a percent. So, even though the % is correct, I would emphasize finding the answer as a fraction in its lowest terms.

One last note, please remember that any fraction can be stated as another equivalent fraction simply by multiplying or dividing the fraction by the number 1, and any fraction where the numerator and denominator are the same (N/N) equals 1.

So, 32/48 divided by 16/16 , still has the value of 32/48, but it is more universally recognized as 2/3. 400/500 = 4/5 not because "you can drop the zeros", but because 400/500 divided by 100/100 = 4/5!

Similarly, 3/4 = 9/12, because 3/4 X 3/3 = 9/12. 5/6 = 10/12, because 5/6 X 2/2 =10/12!

I hope you are not confused, but if you are, please send me your comments :0}