COUNTING

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SIMPLE MATH ON THE ABACUS
COUNTING MULTIPLICATION ADDITION SUBTRACTION
MENU
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COUNTING ON THE PYTHAGOREAN ABACUS
It is always helpful to use a story or image
sequences to direct solution processes when
teaching math. For the process of counting, the
work of gnomes mining and stacking stones from a
mountain may be used.
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In the top figure one gnome sits on a resting
stone. Push the leftmost bead against the left
post.
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Now the gnome, hard at work, brought out a stone
and placed it on the ground. The next white bead
has one brown bead above it.
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Then two gnomes followed and one placed his stone
on the ground beside the first, and the other
placed his stone above and between the two on the
ground. And so the hard working gnomes continued
from 1 to 9.
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SIMPLE ADDITION
A little gnome wanted to be king,So he built a
rocky tower thing.
? 2 4
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He climbed part-way up doing his best.
? 2 4
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Then the next day he climbed the rest.
? 2 4
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NOTE SYMBOL IN GRAPH
6 2 4
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SIMPLE SUBTRACTION
The little gnome started down,But could not
reach the ground.
? 6 - ?
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(count from right 4 beads)
He had climbed to the topmost stone,
? 6 - 4
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But now from the bottom some were gone.
NOTE SYMBOL IN GRAPH
2 6 - 4
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To explore the addition and subtraction facts,
students can solve problems by drawing pictures
of the solutions. These solution can be derived
from the images that motivated exploration on the
abacus. The following slides show the solution
sequence on the abacus corresponding to the
sequence of problems and pictures to the right.
6 1 5
6 2 4
6 3 3
6 4 2
6 5 1
6 6 0
6 7 -1
6 8 - 2
6 9 - 3
6 10 - 4
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6 1 5
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6 2 4
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6 3 3
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6 4 2
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6 5 1
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6 6 0
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6 7 -1
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6 8 - 2
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6 9 - 3
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6 10 - 4
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A hardworking little gnomeEveryday went mining
stones.
MULTIPLICATION
The image sequence of a gnome putting stones from
tunnels into a cart may be used for
multiplication. The gnome takes three stones from
each tunnel and puts them in columns in his cart.
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First he dug deep, deep holesDown to where the
crystals grow
4 ? ?
The abacus shows a triangle with a three-beads
base, pushed to the right.
3
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Then carefully from each He gathered all the
stonesHe could reach.
4 3 ?
The gnome fills his cart with four columns. On
the abacus an adjacent triangle with four beads
in its base is pushed to the right, displacing a
rectangle above it.
3
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NOTE SYMBOL CART MAKES
4 3 12
Pushing this triangle back to the left leaves the
rectangle above it in the middle of the two. The
rectangle includes a total number of beads equal
to the stones the gnome placed in this cart.
4 3
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To explore the multiplication facts students can
solve problems by drawing pictures of the
solutions. These solutions are derived from the
tunnel mining images that motivated exploration
on the abacus. The following slides show that
consecutively pushing to the right then back to
the left bottom beads adjacent to a triangle,
with a base that represents a factor, such as
three in the example, allows the explorations of
a sequence of multiplication facts.
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The columns of beads isolated by the sequence of
manipulations represent the stones taken from the
tunnels and put in the gnomes cart.
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