Informally, the product of multiplying an integer (a "whole" number, positive, negative or zero) times itself is called a square number, or simply "a square." For example, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on, are all squares.

If n is an integer, a square number is n*n or n2.

## Mathematical background

### Objects arranged in a square array

A square number is a set of objects that can be arranged in a rectangle to form a perfect square.

To learn how many pennies (or square tiles) can be arranged in a perfect square array, children can experiment with them.

A penny can:

A penny can:

A sixteen-penny piece can do the following:

This cannot be done with seven pennies or twelve pennies.Numbers (of objects) that form a square array are called "square numbers."

To count as a square number, the square array must be full.As there are 12 pennies here, but they are not arranged in a square array, so 12 is not a square number.

There is no square number 12 in the world.

These kinds of open squares might be of interest to children.The patterns aren't called square numbers, but they are interesting to explore.

It's also fun to make squares out of square tiles.Square numbers are the number of square tiles that can fit into a square array.

Two boards are shown here: 3 by 3 and 5 by 5.What is the percentage of red tiles in each of these?

**Square numbers in the multiplication table**

Square numbers appear along the diagonal of a standard multiplication table.

### Connections with triangular numbers

If you count the green triangles in each of these designs, the sequence of numbers you see is: 1, 3, 6, 10, 15, 21, …, a sequence called (appropriately enough) the triangular numbers.

If you count the white triangles that are in the “spaces” between the green ones, the sequence of numbers starts with 0 (because the first design has no gaps) and then continues: 1, 3, 6, 10, 15, …, again triangular numbers!Remarkably, if you count all the tiny triangles in each design—both green and white—the numbers are square numbers!

### A connection between square and triangular numbers, seen another way

Build a stair-step arrangement of Cuisenaire rods, say W, R, G. Then build the very next stair-step: W, R, G, P.

Each is “triangular” (if we ignore the stepwise edge). Put the two consecutive triangles together, and they make a square:

. This square is the same size as 16 white rods arranged in a square. The number 16 is a square number, “4 squared,” the square of the length of the longest rod (as measured with white rods).Here’s another example:

. When placed together, these make a square whose area is 64, again the square of the length (in white rods) of the longest rod. (The brown rod is 8 white rods long, and 64 is 8 times 8, or “8 squared.”)### Stair steps from square numbers

Stair steps that go up and then back down again, like this, also contain a square number of tiles. When the tiles are checkerboarded, as they are here, an addition sentence that describes the number of red tiles (10), the number of black tiles (6), and the total number of tiles (16) shows, again, the connection between triangular numbers and square numbers: 10 + 6 = 16.

Inviting children in grade 2 (or even 1) to build stair-step patterns and write number sentences that describe these patterns is a nice way to give them practice with descriptive number sentences and also becoming “friends” with square numbers.

Here are two examples. Color is used here to help you see what is being described. Children enjoy color, but don’t need it, and can often see creative ways of describing stair-step patterns that they have built with single-color tiles. Or they might color on 1″ graph paper to record their stair-step pattern, and show how they translated it into a number sentence.A diamond-shape made from pennies can also be described by the 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 number sentence.### From one square number to the next: two images with Cuisenaire rods

(1) Begin with W.Make two successive rods of W+R; then make two more, R+G; then G+P; then..

1; | add 1+2; | add 2+3; | add 3+4; | add 4+5; | add 5+6; | add 6+7 |

W is the starting point for each new square.For each adjacent square, add two rods that match its sides, and add a new W at the corner.