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.
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.
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:
Here’s another example:
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.
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.