Counting and Setting Numbers
Short Course
This is a short course (slide show) in counting and setting numbers. Click on the arrow on the right to begin the slide show.
Counting with Your Fingers
Do you use sometimes use your fingers to count? Sure, many people do (and you may as well) 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]:
Let’s count to 9 using the fingers on your right hand
To start, make sure that none of your fingers are touching a surface
To count from 1 to 4, press each finger to the surface. Don’t press you thumb down yet.
To count 5, press your thumb to the surface and raise the other four fingers in one continuous motion.
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
So, counting to 9, with your right hand, will look like this:
First finger, only, touch the surface.
First finger and second finger touch the surface.
First finger, second finger, and third finger touch the surface.
First finger, second finger, third finger, and fourth finger touch the surface.
Only your thumb touches the surface.
Your thumb and first finger touch the surface.
Your thumb, first finger and second finger touch the surface.
Your thumb, first finger, second finger, and third finger touch the surface.
Your thumb, first finger, second finger, third finger, and fourth finger touch 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 equals 2, three fingers equals 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 equals 9.
Now let’s count to 99 with the fingers on both hands.
Count from 1 to 9 on your right hand.
For 10, clear all your fingers on your right hand press the first finger on your left hand.
For 11, continue to press the first finger on your left and press the first finger on your right hand. Your right hand will provide the unit value of the number and your left hand will keep track of the tens values. Eleven equals 1 ten and 1 unit.
For 12 through 19, continue to press the first finger on the left hand (1 ten) and then press two or more fingers (or the thumb) for each unit.
To count 20, clear all your fingers on your right hand and press the first and second fingers on your left hand.
Continue counting until all four fingers on your left hand are touching the surface and all 4 fingers and your thumb on your right hand are touching the surface as well (4 tens and 9 units).
To count 50, clear all your fingers on your right hand and press your thumb on your left hand to the surface (5 tens and 0 units).
Now you can count the remaining numbers, to 99, by adding units on your right hand and tens on your left hand.
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, pages. 6-88
Counting with an Abacus
We will be using a Cranmer abacus for all of our counting and calculations. What is a Cranmer Abacus and why is it used on this website? This is the abacus that is most often used by Teachers of the Visually Impaired (TVIs) in the United States. The Cranmer Abacus consists of 13 rods with a bar extending across each of the rods. Each rod contains 5 beads, one bead above the bar and four beads below the bar. The major difference between the Cranmer Abacus and the Japanese Soroban is that the Cranmer Abacus has a solid backing which allows the beads to remain stationary. This is an important advantage to a person who has limited or no sight. Should their Cranmer Abacus be dropped, the number value that was entered will still be displayed. The Cranmer Abacus that can be purchased from the American Printing House for the Blind has raised bumps at the bottom of each rod and on the bar. After every three rods, you will also feel (“see”) a larger bump or marker. The raised parts are particularly helpful to individuals with limited sight. The markers will be used in future calculations, particularly for decimals and fractions. For more information on the Cranmer abacus refer to the Short History of Numbers and the Abacus document.
Now, let's make the connection between counting with your fingers and counting with an abacus.
Place Value
Each column of the abacus represents a digit in a number. The most significant digit (the digit with the highest value) is the left most digit of the number. The place value document shows how each column in the abacus corresponds to each digit of a number.
Setting and Clearing Numbers
As we discussed earlier, the most basic way to enter a number into our Cranmer abacus is by pushing one bead at a time to the bar. Eventually we would arrive at the number we wanted to enter.
However, a faster way to enter number is to push groups of beads to the bar. We call this setting numbers directly. When we use the place value of each column, we can enter numbers quickly. Using a few simple steps and the place value of each column discussed above, we can set any number up to and including 9,999,999,999,999 (since our Cranmer abacus had 13 columns). These are the steps you need to follow:
Set the numbers 1 through 4, by pushing groups of beads from below the bar, to the bar
Set the number 5, by pushing the 5 bead to the bar.
Set the numbers 6 through 9, by pushing (or pinching) the 5 bead and groups of beads from below the bar, to the bar
Clear any number by pulling beads away from the bar
Examples
Let's look at some examples of setting and clearing numbers:
Example: Set 1
Example: Set 2
Example: Set 3
Example: Set 4
Example: Set 5
Example: Set 6
Example: Set 7
Example: Set 8
Example: Set 9
Example: Set 10
Example: Set 27
Example: Set 1,791
Example: Set 483,349
Example: Set 20,030,327