Informally: once you multiply an essence (a “whole” number, positive, negative or zero) time itself, the resulting product is called a square number, or a perfect square or merely “a square.” So, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on, space all square numbers.

You are watching: 1+4+9+16+25

More formally: A square number is a number of the kind n × n or n2 wherein n is any integer.

Mathematical background

Objects i ordered it in a square array

The surname “square number” comes from the fact that these particular numbers of objects deserve to be i ordered it to to fill a perfect square.

Children have the right to experiment v pennies (or square tiles) to check out what numbers of them can be i ordered it in a perfect square array.

Four pennies can: 

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Nine pennies can: 

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And sixteen pennies can, too: 

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But seven pennies or twelve pennies cannot be arranged the way. Number (of objects) that can be arranged right into a square array are called “square numbers.

Square arrays need to be full if we room to count the number as a square number. Here, 12 pennies are arranged in a square, yet not a complete square array, therefore 12 is not a square number.


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The number 12 is not a square number.


Children might enjoy exploring what number of pennies deserve to be arranged into an open square favor this. They space not called “square numbers” however do monitor an exciting pattern.

Squares make of square tiles are also fun come make. The number of square tiles the fit into a square variety is a “square number.”


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Here space two boards, 3 × 3 and also 5 × 5. How countless red tiles in each? Black? Yellow?Are any of those square numbers?What if you brick a 4 × 4 or 6 × 6 board the very same way?Can you predict the variety of tiles in a 7 × 7 or 10 × 10 board?

Square numbers in the multiplication table


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Square numbers show up along the diagonal of a typical multiplication table.


Connections v triangular numbers

If you counting the green triangles in every of this designs, the succession of numbers you check out is: 1, 3, 6, 10, 15, 21, …, a sequence referred to as (appropriately enough) the triangle numbers.

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If you count the white triangle that are in the “spaces” between the green ones, the sequence of number starts with 0 (because the an initial design has no gaps) and then continues: 1, 3, 6, 10, 15, …, again triangle numbers!

Remarkably, if you counting all the small triangles in each design—both green and also white—the numbers space square numbers!

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

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

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Each is “triangular” (if we overlook the stepwise edge). Put the 2 consecutive triangles together, and also they do a square:

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. This square is the very same size together 16 white rods i ordered it in a square. The number 16 is a square number, “4 squared,” the square that the size of the longest rod (as measured through white rods).

Here’s an additional example:

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. When put together, these make a square
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who area is 64, again the square that the length (in white rods) the the longest rod. (The brown rod is 8 white rods long, and also 64 is 8 time 8, or “8 squared.”)

Stair actions from square numbers

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Stair actions that go up and also then earlier down again, favor this, additionally contain a square number of tiles. As soon as the tiles room checkerboarded, together they are here, an enhancement sentence that describes the variety of red tiles (10), the number of black tiles (6), and the total variety of tiles (16) shows, again, the connection in between triangular numbers and square numbers: 10 + 6 = 16.

Inviting kids in class 2 (or even 1) to build stair-step patterns and also write number sentences that describe these trends is a nice method to provide them exercise with descriptive number sentence and likewise becoming “friends” v square numbers.

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Here are two examples. Color is used right here to help you see what is being described. Children enjoy color, but don’t require it, and also can regularly see an imaginative ways of describing stair-step trends that castle have constructed with single-color tiles. Or lock might shade on 1″ graph document to document their stair-step pattern, and show just how they interpreted it into a number sentence.
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A diamond-shape do from pennies can also be described by the 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 number sentence.

From one square number come the next: two photos with Cuisenaire rods

(1) start with W. Include two continually rods, W+R; then an additional two, R+G; climate G+P; then….

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

(2) begin with W. Because that each new square, include two rods that enhance the political parties of the vault square, and also a new W to fill the corner.