Q:

Concrete way to teach addition and subtraction of fractions

I am teaching 4th-grade kids. The topic is Fraction. Basic understanding of a fraction as a part of the whole and as part of the collection is clear to the kids. Several concrete ways exist to teach this basic concept. But when it comes to fraction addition/subtraction I could not find a way that teaches it concretely.
Of course, teaching fraction addition & subtraction of the form 3/2 + 1/2 is easy. But what about 3/2+ 4/3?
It is where we start talking about the algorithm (using LCM), which makes the matter less intuitive and more abstract which I am trying to avoid in the beginning. I believe all abstract concepts should come after the concrete experience.   
So teachers do you have any suggestions? 

A:

Use a piece of paper as your whole.   To teach $3/2 + 4/3$ do the following.

Give each child/group of children 6 pieces of paper.
One piece of  paper should be left as a whole - the students can
write 1 whole on the paper.
Have the students fold 2 of the pieces in half lengthwise, label
each half as $1/2$, and cut out the halves.
Have the students fold 2 pieces of papers in thirds widthwise, label
each third as $1/3$ and cut out the thirds.
Have the students take $3/2$ and $4/3$ and try to add them. You will
have 2 wholes and $1/2$ and $1/3$  Discuss  what to do with the
leftover pieces...
Next take the last piece of paper and fold in sixths by folding in
half lengthwise and in thirds widthwise.  Label each piece $1/6$ and
cut out the pieces.  
Put the some of the sixths from part 6 on the leftover pieces in
part 5 to show that $1/2=3/6$ and $1/3 = 2/6$.  Together they make
$5/6$

Hope this works for you.

A:

I like the cutting paper approach the best because it is cheap and easy, and it forces students to confront basic fractions by folding/cutting them out. But here’s another thought...
For you to demonstrate: have several clear-plastic, cylindrical cups, each marked off in different fractions of a whole (maybe start with 1/2’s, 1/3’s, 1/4’s, 1/6’s, 1/8’s, 1/10’s and 1/12’s). Have another larger cylinder marked off in whole numbers of the smaller cups.
Write the problem $\frac{3}{2} + \frac{4}{3}$ and have students make a guess/estimate, and then use the 1/2 and 1/3 cups to fill up with colored liquid or rice (which is why you will demonstrate this). Add the liquid to the large cylinder to see that “yeah, it’s between 2 and 3”, as the students should have guessed. Then work backward, pouring away whole numbers worth of smaller cups until you have less than a whole cup left In the large cylinder. Now try the smaller cups one-by-one until you find one in which the remaining liquid fills up to a marker line. Finally, declare the answer as the mixed number $2 \frac{5}{6}$. 
I might do this with two nice examples (where I have the appropriately-marked cups) and then try the first one out again, but pour each of the original fractions into the 1/6’s cup before adding them together in the large cylinder. Lots of things you could do with these. 

A:

Suggestion: Do not bother having students do calculations with two fractions until they can do a comparable operation with a fraction and a whole number.
Example: Before teaching, say, $\frac{3}{2}-\frac{4}{3}=\_\_$, ensure they can answer something like $4 - \frac{1}{5}=\_\_$. Make sure all students can draw that on a number line and with area diagrams. This is how you ensure that their knowledge of fractions will integrate with their knowledge of whole numbers.
Once students are ready to add and subtract fractions that already have common denominators, ensure they can do so on a number line and with area diagrams and with fraction tiles.
Then, when it comes time to do $\frac{3}{2}+\frac{4}{3}=\_\_$, you can:

Put the halves tiles and the thirds tiles together
Pretend to be horribly confused by how much that is
Estimate it to be a little under 3 wholes
Demonstrate how sixths tiles come to the rescue.   :)

