1. SIMPLE MATH ON THE ABACUS
COUNTING
MULTIPLICATION
ADDITION
SUBTRACTION

2. 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.

3. In the top figure one gnome sits on a resting stone.
Push the leftmost bead against the left post.

4. 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.

5. 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.
Space between last two lines of first sentence

6. SIMPLE ADDITION
A little gnome went mining stones,
? = 2 + 4

7. He dugout 2 before snack.
? = 2 + 4

8. Then he dugout 4 more for his sack.
? = 2 + 4

9. NOTE SYMBOL IN GRAPH
6 = 2 + 4

10. SIMPLE SUBTRACTION
The little gnome went mining stones,

11. (count from right 4 beads)
He dugout 6 stones before snack,
? = 6 - ?

12. But he dropped 4 from his sack.
NOTE SYMBOL IN GRAPH
? = 6 - 4

13. 2 = 6 - 4 So there were only 2 left upon his back.
NOTE SYMBOL IN GRAPH
2 = 6 - 4

14. 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
Rework first sentence. What is derived, the pictures or the solutions?
6 = 9 - 3
6 =

15. 6 = 1 + 5

16. 6 = 2 + 4

17. 6 = 3 + 3

18. 6 = 4 + 2

19. 6 = 5 + 1

20. 6 = 6 + 0

21. 6 = 7 -1

22. 6 = 8 - 2

23. 6 = 9 - 3

24. 6 =

25. A hardworking little gnome Everyday 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.

26. First he dug deep, deep holes Down to where the crystals grow
4 × ? = ?
The abacus shows a triangle with a three-beads base, pushed to the right.
How about—The abacus shows a triangle with a three-bead base, pushed to the right.
× 3

27. Then carefully from each He gathered all the stones He could reach.
4 × = ?
The gnome fills his cart with four columns. On the abacus an adjacent triangle with four beads in it’s base is pushed to the right, displacing a rectangle above it.
× 3

28. NOTE SYMBOL CART MAKES
4 × = 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. ×

29. 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.
1 X 3
2 X 3
3 X 3
4 X 3
5 X 3
Rework these sentences.
6 X 3
7 X 3

30. 1 X 3
The columns of beads isolated by the sequence of manipulations represent the stones taken from the tunnels and put in the gnome’s cart.

31. 2 X 3

32. 3 X 3

33. 4 X 3

34. 5 X 3

35. 6 X 3

36. 7 X 3