![]() 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. Stair steps that go up and then back down again, like this, also contain a square number of tiles. (The brown rod is 8 white rods long, and 64 is 8 times 8, or “8 squared.”) Stair steps from square numbers 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 number 16 is a square number, “4 squared,” the square of the length of the longest rod (as measured with white rods). This square is the same size as 16 white rods arranged in a square. ![]() Put the two consecutive triangles together, and they make a square. Then build the very next stair-step: W, R, G, P.Įach is “triangular” (if we ignore the stepwise edge). 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 wayīuild a stair-step arrangement of Cuisenaire rods, say W, R, G. 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! ![]() 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. Square numbers appear along the diagonal of a standard multiplication table. Here, 12 pennies are arranged in a square, but not a full square array, so 12 is not a square number. Square arrays must be full if we are to count the number as a square number. Numbers (of objects) that can be arranged into a square array are called “square numbers. The name “square number” comes from the fact that these particular numbers of objects can be arranged to fill a perfect square.Ĭhildren can experiment with pennies (or square tiles) to see what numbers of them can be arranged in a perfectly square array.īut seven pennies or twelve pennies cannot be arranged that way. Mathematical background Objects arranged in a square array More formally: A square number is a number of the form n × n or n 2 where n is any integer. Informally: When you multiply an integer (a “whole” number, positive, negative or zero) times itself, the resulting product is called a square number, or a perfect square or simply “a square.” So, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on, are all square numbers.
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