Our numbering system has two basic characteristics.

First of all, it is a **positional system** in which the same number has a different value depending on the position it occupies. Look at 4, for example. In 2 **4** , it is worth **4** units. In the **4** 5, however, it is worth 4 tens (that is, **40** ). And in the **4** 13, 4 hundreds (or **400** ).

This fact differentiates our numbering system from Roman numerals, for example, which is based on an additive system, in which the same letter always has the same value. Although, over time, this system also incorporated some positional element to reduce the number of digits necessary, and a letter placed in front of another of greater value has a negative value. The I, for example, is worth -1 if it is in front of the V (IV = 4) and +1 if it is behind (VI = 6).

Secondly, it is a **decimal or base 10 system** , which means that quantities are grouped in tens: 10 ones make a ten, 10 tens make a hundred, and so on. Thus, 10 digits (from 0 to 9) are needed to indicate how many units, how many tens, how many hundreds, etc., a certain number has.

There are other positional systems that use other bases, such as the binary system (base 2), which is used especially in the computer field and in which there are only two figures, 0 and 1.

Understanding how our number system works is very important to perform arithmetic operations.

And one of the basic skills required for this is **numerical decomposition** , specifically:

- The decomposition of a number into two or more quantities.

- The decomposition of a number into the number of units, tens, hundreds, etc. containing. For example: 235 = 200 + 30 + 5.

Below I am going to present 3 ideas to work on numerical decomposition up to 10 with children between 5 and 7 years old, although you can also propose them with older children, especially if they need to develop mental calculation strategies.

## How to decompose with a ball chain

A chain with balls is a basic material for decomposing numbers. They come in different lengths (the most common are 10, 20 or 100 balls), with wooden or plastic balls and in two different colors, generally white and red, to facilitate counting.

I recommend starting with a chain of 10 balls, five red and five white. If you have a longer chain, you can simply undo the knot at one end, remove the excess balls, and tie the knot again.

In ball chains, the string is long enough to separate the balls into different groups.

Ask the boys and girls to separate them into two groups (they can use a clamp or something similar to mark the separation) and to count how many balls there are in each of them. This is called breaking down the number 10, and there are many different ways to do it. You can encourage them to find more than one way (or all the possible ones), and record the results obtained, either graphically or with numbers.

When counting the balls, look at the counting strategies that each one uses:

Are they able to identify small quantities without having to count the balls one by one? Do they count from 5 (which is equivalent to all balls of the same color) when there are a greater number of balls? Knowing the number of balls in one of the groups, can you deduce the other?

*Note* : you can make this material at home, with beads, macaroni or any other element that can be strung.

## Numerical decomposition and polycubes

Polycubes are plastic cubes with a 2 cm edge and 10 different **colors** that can fit together. In addition to being excellent play material, they can be used for endless mathematical activities.

Propose the boys and girls to build towers of 10 polycubes and then split them in two:

How many different ways can they do it?

Then encourage them to draw on graph paper the different results they have obtained and express them numerically, as in the example in the following image.

Another possibility is to build towers of 10 polycubes using two different colors. How much do you need of each color?

*Note* : If you don’t have polycubes, you can propose this activity with Lego-type pieces.

## Decomposing the number 10 with rulers

Cuisenaire rulers are wooden bars with a square section (1 cm x 1 cm) that represent the first ten natural numbers.

The strip that represents the number 1 (or unit strip) measures 1 cm in length. The one that represents the number 2, which is equivalent to two unit strips, measures 2 cm. The one that represents 3.3 cm. And so on until you reach the strip that represents the number 10, which is equivalent to 10 unit strips and measures 10 cm.

Each of these strips has a different color to be able to identify them more easily.

One of the classic activities carried out with the strips is precisely the numerical decomposition of 10. That is, looking for two strips that, placed one after the other, are equivalent to the number 10 strip (or, what is the same, have the same length).

Propose to the boys and girls that they find all the possible combinations and that they build with them what is called the *wall of strips* of number 10.

The results obtained can be recorded first graphically, using graph paper, and then numerically, as you see in the image.

You can build the wall of any other number, just take the corresponding strip as a reference. And you can also use more than 2 power strips per row.

*Observation:* a homemade way to do these activities is to build cardboard strips. The unit strip does not have to be 1 cm x 1 cm, you can make it 2 cm x 2 cm and it will be easier to manipulate.