1. FRACTIONS ON THE PYTHABACUS
SHORT DIVISION
FRACTIONS x WHOLES
MULTIPLICATION
DIVISION
REDUCING

2. FRACTIONS
SHORT DIVISION To begin the fraction lesson teach short division on the abacus. I recommend using a story or sequence of images to direct the solution process for students. I might tell the students that a family of three squirrels searched for nuts on an autumn morning and found fourteen nuts. They each took one nut in turn from their collection until there were not enough left for them each to have another. They left the extra for the winter birds. How many nuts did each squirrel get.

3. PUSH TRIANGLE REMAINING ON THE LEFT TO MIDDLE OF ABACUS.
Push to the right a triangle with beads in its base equal the number of squirrels, the divisor Then push from the brown columns to the right above the divisor a number of beads equal the nuts collected by the squirrels, the dividend.

4. The abacus now shows that the triangle remaining on the left side has a base number of beads equal the number of nuts each squirrel gets, the whole number part of the quotient. The last column of beads, of the dividend, equals what remained for the winter birds.

5. 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 cry But, 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.

6. 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 cry But, 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.

7. THREE-FOURTHS OF FOUR CORRESPONDS TO THREE-FOURTHS OF TWELVE
To begin the solution process have the students multiply the denominators, to see into how many pieces the pie is sliced. This product, twelve, is the denominator of the solution. Direct students to write it under the fraction bar.

8. TWO-THIRD 0F THREE CORRESPONDS TO TWO-THIRDS OF NINE
Now to find the numerator students must figure out how many pieces of the whole pie the friend ate. Have students first find how many pieces remained when the friend arrived, by taking three-fourths of twelve, the whole pie. The answer is nine.

9. Then students can find out how much of the three- fourths or nine pieces the friend ate, by taking two- thirds of the nine. As shown above. the answer is six Direct students to write six above the fraction bar. 

10. DIVIDING FRACTIONS
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.

11. THREE-FOURTHS 0F FOUR CORRESPONDS TO THREE-FOURTHS OF TWELVE
Have the students, as before, multiply the denominators to see into how many equal pieces the pie is sliced (twelve), and position the quadrilateral of beads between the triangles. Then multiply the second fraction times twelve, the number of slices in the whole pie.

12. TWO-THIRD 0F THREE CORRESPONDS TO TWO-THIRDS OF TWELVE
This fraction of beads, nine, is represented to students as the slices eaten of the first pie. It is the denominator of the solution fraction. Have students write it under the fraction bar. Now, have students multiply the first fraction times twelve, the slices in the whole pie.

13. 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 students write it over the fraction bar. 

16. 2 4 1 3 5
If the numerator (4) and denominator (6) or represented as a square and a hexagon the beads of the abacus may represent the points at the vertices of these polygons. Now from a beginning point or vertex, count clockwise around the polygons to select points separated by a number equal the difference between the numerator and denominator, in this case (2) . The number of selected points for each polygon will be less than the number of points of the given polygons by a common factor. The triangular array of the abacus coordinates this property of polygons.