Thinking Fractions

If you think about what you are doing when working with fractions everything becomes crystal clear. The worst thing to do is to visualize the algorithm you learned in elementary school; something like 1/2 + 2/3 or 1/4 x 4/5 or, heaven forbid, 1/2 divided by 1/4. If we do this then we are confined to thinking instrumentally procedurally about something that is comprised of only procedural knowledge. We fall back on senseless rules like "find the common denominator" or "cross multiply" or "change the sign and flip the second fraction". learning tricks like this will probably get you a better score on a traditional math test (like those used to compile the TIMMS report)  but will do little to help you understand what fractions are all about.

So for 1/2 + 2/3 think about different ways you could make 1/2. You could say it's the same as 2/4, 3/6, 4/8 and so on. Now do the same with 2/3. It could be 4/6. Aha, no need to go any further if you know that you can add fractions that have the same denominator. So 3 sixths plus 4 sixths is 7 sixths. We can count sixths just like anything else. Now you can see how you can get sixths by multiplying 2 by 3 if you need to do more difficult ones. But at least you now understand why.

1/4 x 4/5 is more challenging because the x (multiplication symbol) and the whole concept of multiplication of fractions can be different from using it with whole numbers. Visualize 4/5 using the model in the picture above. Now take 1/4 of that 4/5 and you end up with 1/5. The key to understanding this is seeing how the size of the 1 to which each fraction refers changes. The 1 of the 4/5 is the red 1 above whereas the 1 being referred to by the 1/4 is the 4/5. Really we're asking what is 1/4 of 4/5? Once you start to see the pattern, the relationship between the numerator and denominator then things get easier. Try 1/3 x 3/5 or 1/2 x 2/7 or 1/4 x 4/9. If there is not relationship between the numerator and denominator you can do it another way. For example 1/2 x 3/8 is 3/16. This can be done conceptually by halving the size of the fractional pieces. 1/16 is half of 1/8 so 2/16 is 1/2 of 3/8.

1/2 divided by 1/4 is really asking how many 1/4s are there in a 1/2. This is easily conceptualized by thinking about a football game; how many 1/4s are there in the first 1/2? Clearly there are 2.

3/8 divided by 1/4 is a little more difficult because the referent of the fractions changes. Again, think how many 1/4s are there in 3/8 by visualizing the fractional pieces. Compare 1/4 (a yellow piece above) with 3/8 (three dark blue pieces and you'll see there are 1 1/2 quarters in 3/8. The referent for the 1 1/2 is the 1/4.

from this conceptual understanding it's easier to "mathematize", to use Bob Wright's term,  what is going on by working out how the procedural knowledge of operating with fractions works.