From Fingers to Frames

Seeing smaller groups of objects within larger groups helps us determine the total number of objects more efficiently than counting each object one by one. Sometimes our brains “see” smaller groups inside of larger groups like in the group of nine dots below.  

Two Hands and Ten Fingers

Early in our mathematical journey, we learn that small numbers can be combined to make larger numbers, and that large numbers can be broken up into combinations of smaller numbers.  To help us learn to compose and decompose numbers, we can access a common math tool, our hands!  With our fingers we can show quantities up to 10.

With 10 fingers on 2 hands, we can count to 10 using one-to-one correspondence, or we can identify numbers up to 10 using different finger combinations. For instance, we “see” that 8 can be a group of5 and 3 or4 and 4.

Anthropologists believe that humans settled on the base ten number system, in large part, because of our 10 fingers.

This begs the question, would we have a different number system if we had 12 fingers? 

Number Rack

Number racks, sometimes called rekenreks, are used to represent quantities and combinations in rows of 10 beads, each row containing 5 red and 5 white beads. How might this model relate to our fingers and hands?

• An abacus or a number rack?

These two counting devices look similar, but the value of individual beads on an abacus changes depending on the bead’s location. On a number rack, a bead is worth 1 unit regardless of its position or row. The number rack also uses two colors—red and white—to focus attention on groups of 5. The abacus is an ancient tool and is used around the world still today. To learn more about its modern day use, reference Learn More below.

A number rack starts with all its beads on the right, showing zero on the left. To represent quantity, beads are moved to the left side, or to the "counting area."

Slide beads below to the left, one by one or in groups, to show the number 6.

What are the different quantities of beads that can be combined to make 6?

Number racks commonly have 2 rows, each with 10 beads. By using 2 rows, one can show various ways to combine smaller numbers to compose larger numbers up to 20.

How might this number rack be used to show 16? 

Perhaps you thought of 16 as 10 + 6 or 8 + 8. Is 9 + 7 a valid combination for 16?

10 + 6 shows 16 using five-wise counting: 5 + 5 + 5 + 1.

8 + 8 shows 16 using pair-wise counting: 8 pairs of beads.

Nine on the top and 7 on the bottom also make 16 total beads.

Number Frames

Frames are another visual tool used to  group objects for counting. Here are three empty frames: a five-, ten-, and double-ten-frame (twenty-frame).

Placing counters inside the frames helps us represent quantity with recognized number groups.

Frames support the ability to subitize, or determine quantity at a glance. See how many quantities you can identify in the following frames. Press “Show” to start. Be ready because each frame appears for only a second. 

• More About Zero

The previous chapter discussed why zero is not easily represented with dots. 

Without context, the rectangle is only a rectangle—not the best representation for a set of zero objects.

Hands, number racks, and number frames provide helpful contexts to represent zero. They each convey empty sets with no quantity.

Grouping numbers into smaller, identifiable numbers of visual objects—our fingers, number rack beads, or counters in number frames—helps us determine quantity more efficiently than counting one by one. Visually identifying small sets of numbers within larger numbers is called conceptual subitizing. Recognizing groups of 5 and 10 is a key skill that will help us in our mathematical journey. It sets the stage for representing and identifying larger numbers.