I invited over a friend, baked a pie,Then, decided a little piece I'd try.In an hour my friend arrived.What remained made them cryBut, just a bit they ate with a sigh.So all alone, I finished the pie. 

Once students can multiply a whole number by a fraction on the abacus, a sequence of manipulations can easily be learned to solve multiplication and division of fraction problems. It is again helpful to direct the solution process with a story or image sequence. The poem above may be the bases for such a sequence.

MULTIPLICATION OF FRACTIONS

I invited over a friend, baked a pie,Then, decided a little piece I'd try.In an hour my friend arrived.What remained made them cryBut, just a bit they ate with a sigh.So all alone, I finished the pie. 

In the fraction problem shown below, the second fraction, reading from left to right, is how much of the pie remained when the friend arrived, and the first fraction is how much of what remained the friend ate.

To begin the solution process have the students multiply the denominators, as shown on abacus A, to see into how many coins the pie is sliced. This product, twelve, is the denominator of the solution. Direct students to write it under the fraction bar of the solution fraction.

THREE-FOURTHS OF FOUR CORRESPONDS TOTHREE-FOURTHS OF TWELVE

Now to find the numerator students must figure out how many coins of the whole pie the friend ate. Have students first find how many coins remained when the friend arrived, by taking three-fourths of twelve, the whole pie. As shown over Abacus B and abacus C, the answer is nine.

TWO-THIRD0F THREECORRESPONDS TOTWO-THIRDSOF NINE

Then students can find out how much of the three-fourths or nine coins the friend ate, by taking two-thirds of the nine coins. As shown over abacus D and abacus E the answer is six. Direct students to write six above the fraction bar of the solution fraction 

I invited again my friend for pie.Have no fear this is why;For - I baked two with pride.I ate most of one, but no need to cry.Here's another for my friend to try.Eat my friend and don't be shy.

Division of fractions can be shown to be a comparison of one fraction to another. The solution process can be directed by continuing our poem.

DIVIDING FRACTION

Have the students, as before, multiply the denominators to see into how many equal coins the pie is sliced (twelve), but position the quadrilateral of beads between the triangles, as shown on abacus A. Now they are prepared to take a fraction of the pie.

BC

Have students multiply the second fraction times twelve, the number of slices. As shown over abacus B, abacus C and abacus D the answer is nine. This fraction of beads is represented to students as the slices eaten of the first pie and is the denominator of the solution fraction. Have students write it under the solution fraction bar.

DTHREE-FOURTHSOF FOURCORRESPONDS TOTHREE-FOURTHSOF TWELVE

EFG

Now, have students multiply the first fraction times twelve, the products of the denominators. As shown over abacus E, abacus F and abacus G the answer is eight. This fraction of beads is represented to the students as the slices eaten of the second pie by the friend and is the numerator of the solution fraction. Have student write it over the solution fraction bar. 

TWO-THIRDS OF THREECORRESPONDS TOTWO-THIRDSOF TWELVE