MULTIPLYING FRACTIONS
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Pythabacus Videos		 Once students can can multiply a whole number by a fraction on the Pythabacus, 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 below may be the bases for such a sequence
I invited over a friend, baked a pie,
Then decided a little piece I'd try.
After 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 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, to see into how many pieces the pie is sliced. So push 3 bottom beads against the left post to represent the denominator of 2/3. 

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Next push 4 more bottom beads a little to the left to represent the denominator of 3/4. Touching the triangle above these four beads there will be a quadrilateral array of 12 brown beads. Therefore 12 will be the denominator of the solution, the whole pie. Direct students to write it under the fraction bar of the solution fraction. 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. Push 3 of the 4 bottom beads to the left against the 3 beads against the left post to represent the numerator of 3/4. Push these 3 beads back to the right to rejoin the 1 remaining to the right. A quadrilateral of 9 brown beads will remain touching the triangle above the 3 beads against the left post.

Then students can find out how much of the three-fourths or nine pieces the friend ate, by taking two-thirds of the nine pieces. Push 2 of the 3 bottom beads, against the left post, to the right midway to the 4 beads of 3/4 to represent the numerator of 2/3. When you push these 2 beads back to the left 6 brown beads will be left in the middle representing the numerator of the solution.
Therefore 2/3 X 3/4 = 6/12.  Direct students to write six above the fraction bar of the solution fraction. 
The process shows to the students that a piece (two-thirds) of a part (three-fourths) of a whole (twelve) is compared to the whole (twelve) to find the answer when multiplying fractions. After solving a number of problems on the Pythabacus students can be guided to discover the numerical rule for multiplying fractions.