The 9 to 10 Exchange

Counting to 10 with your fingers

In the 4 to 5 Exchange section, we counted to 5 using our fingers. In this section, let’s go a little further and count to 10.

1. To start, make sure that none of your fingers are touching a surface

2. To count from 1 to 4, press each finger to the surface. Don’t press you thumb down yet.

3. To count 5, press your thumb to the surface and raise the other four fingers in one    continuous motion.

4. Now, to count from 6 to 9, keep your thumb on the surface and press each finger, in  order, until the remaining 4 fingers are touching the surface

5. For 10, clear all your fingers on your right hand press the first finger on your left hand. 

So, counting to 10, with your right hand and left hand, will look like this:

1. 1. First finger, only, touch the surface.

   2. First finger and second finger touch the surface.

   3. First finger, second finger, and third finger touch the surface.

   4. First finger, second finger, third finger, and fourth finger touch the surface.

   5. Only your thumb touches the surface.

   6. Your thumb and first finger touch the surface.

   7. Your thumb, first finger and second finger touch the surface.

   8. Your thumb, first finger, second finger, and third finger touch the surface.

   9. Your thumb, first finger, second finger, third finger, and fourth finger touch the  surface.

   10. Touch the surfaces with the first finger of your left hand and clear all the fingers in your right hand.

Why does this work?  Each finger has a value of 1 and the thumb has a value of 5. So, the value of each number is increased by 1 as we press the finger to the surface. One finger equals 1, two fingers equal 2, three fingers equal 3 and four fingers equals 4.  The thumb has a value of 5 so pressing just the thumb to the surface equals 5.  Pressing the thumb and one finger (5 + 1) equals 6. The thumb and 2 fingers (5 + 2) equal 7. The thumb and 3 fingers (5 + 3) equal 8 and the thumb and 4 fingers equal 9. The fingers on your left had have a value of 10 and the left thumb has a value of 50.  

Counting to 10 with the Abacus

How do we count 10 with an abacus?  In each step, we push 1 bead to the bar.  Then when all the beads are used below the bar, we push the 5 bead to the bar and clear the 4 beads below the bar.  To count from 9 to 10, we push 1 bead to the bar in the tens column and clear all of the beads in the ones column.

Adding using the 9 to 10 Exchange

As you saw in the previous section, counting to 5 involves using all the beads below the bar and the 5-bead.  Now, let’s add some single numbers and use the 9 to 10 exchange: 

• Add:  9 + 1

• Add: 8 + 2

• Add:  7 + 3

• Add:  6 + 4

Examples

Now you are ready to apply this exchange with some examples with larger numbers. These examples we will use the 9 to 10 exchange in any column when no beads are available below the bar.  Here are some guidelines to follow: 

• Begin counting with the beads below the bar.

• Count one when we push each bead to the bar.

• Never count when you clear.

• Clear all four beads simultaneously and continue counting.

Also, in these examples, each addition calculation will start in the left most column and continue in each column, moving to the right.

• Example:  33 + 98

• Example:  47  + 75

• Example:  24 + 96

• Example:  123 + 987

• Example:  987 + 232

Build Your Skill

Now you are ready to try some calculations on your own. Click on this link for some problems and their solutions to test your skill.