The 4 to 5 Exchange

Counting to 5 with your fingers

Do you use sometimes use your fingers to count?  Sure, many people do even though you were probably taught in school not to use them.  Despite what you were you taught there is nothing wrong with this practice.

If you have all your fingers in good working condition, on both hands, you have a total of 10 fingers. Five fingers on your right hand and 5 fingers on your left hand.  Using a special method of finger counting[1] you can count to 99 using both hands not using American Sign Language or any other type of sign language. You don’t believe me, do you?  Well, here’s how you do it[2]:

However, in this chapter, we will start by only counting to 5 using the fingers on your right hand

1. 1. 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.

So, counting to 5, with your right hand, will look like this:

                 4.      First finger, only, touch the surface.

                 5.      First finger and second finger touch the surface.

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

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

                 8.      Only your thumb touches the surface.

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 equal 4.  The thumb has a value of 5 so pressing just the thumb to the surface equals 5. 

But what does counting with your fingers have to do with learning to do calculations with the Cranmer Abacus?  Actually, counting with your hands and counting numbers with the Cranmer Abacus are very, very similar.

[1] This method of finger counting is from the Fingermath® International School Program

[2] Lieberthal and Lieberthal, Fingermath® International School Programpages. 6-88

Counting to 5 with the Abacus

How do we count with an abacus? Let’s look at how we count to 5.  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. 

Adding using the 4 to 5 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 4 to 5 exchange: 

• Add:  4 + 1 

• Add:  3 + 2

• Add:  2 + 3

• Add:  1 + 4

Examples

Now you are ready to apply this exchange with some examples with larger numbers. These examples we will use the 4 to 5 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:  4 + 3

• Example:  23 + 13

• Example:  26 + 63

• Example:  274 + 321

• Example:  464 + 323

• Example:  2,148 + 6,131

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.