Informally: When you multiply an integer (a “whole” number, positive, negative or zero) times itself, the resulting product is referred to as 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 also so on, are all square numbers.

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More formally: A square number is a variety of the develop n × n or n2 wright here n is any integer.

Mathematical background

Objects arranged in a square array

The name “square number” comes from the truth that these specific numbers of objects deserve to be arranged to fill a perfect square.

Children deserve to experiment via pennies (or square tiles) to view what numbers of them have the right to be arranged in a perfectly square range.

Four pennies can: 

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Nine pennies can: 

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And sixteen pennies can, too: 

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But salso pennies or twelve pennies cannot be arranged that way. Numbers (of objects) that deserve to be arranged into a square variety are dubbed “square numbers.

Square arrays should be complete if we are to count the number as a square number. Here, 12 pennies are arranged in a square, however not a complete square selection, so 12 is not a square number.


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The number 12 is not a square number.


Children might reap trying out what numbers of pennies can be arranged right into an open square choose this. They are not called “square numbers” however execute follow an exciting pattern.

Squares made of square tiles are additionally fun to make. The number of square tiles that fit into a square selection is a “square number.”


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Here are 2 boards, 3 × 3 and 5 × 5. How many red tiles in each? Black? Yellow?Are any of those square numbers?What if you tile a 4 × 4 or 6 × 6 board the very same way?Can you predict the number of tiles in a 7 × 7 or 10 × 10 board?

Square numbers in the multiplication table


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


Connections via triangular numbers

If you count the green triangles in each of these designs, the sequence of numbers you check out is: 1, 3, 6, 10, 15, 21, …, a sequence referred to as (as necessary enough) the triangular numbers.

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If you count the white triangles that are in the “spaces” in between the green ones, the sequence of numbers starts via 0 (bereason the first architecture has actually no gaps) and also then continues: 1, 3, 6, 10, 15, …, aget triangular numbers!

Remarkably, if you count all the tiny triangles in each design—both green and also white—the numbers are square numbers!

A connection in between square and also triangular numbers, watched another way

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

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

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. 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 size of the longest rod (as measured through white rods).

Here’s another example:

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. When put together, these make a square
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whose location is 64, aget the square of the length (in white rods) of the longest rod. (The brvery own rod is 8 white rods long, and also 64 is 8 times 8, or “8 squared.”)

Stair actions from square numbers

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Stair measures that go up and also then earlier down aacquire, choose this, additionally contain a square number of tiles. When the tiles are checkerboarded, as they are here, an addition sentence that describes the variety of red tiles (10), the variety of babsence tiles (6), and also the complete number of tiles (16) shows, again, the link in between triangular numbers and also square numbers: 10 + 6 = 16.

Inviting kids in grade 2 (or even 1) to construct stair-action fads and create number sentences that define these fads is a nice way to offer them practice via descriptive number sentences and additionally ending up being “friends” via square numbers.

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Here are 2 examples. Color is supplied below to assist you view what is being described. Children gain shade, however don’t require it, and also have the right to frequently see imaginative ways of describing stair-action patterns that they have actually developed via single-color tiles. Or they might color on 1″ graph paper to document their stair-step pattern, and also show exactly how they analyzed it into a number sentence.
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A diamond-shape made from pennies have the right to additionally be described by the 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 number sentence.

From one square number to the next: 2 imperiods with Cuisenaire rods

(1) Start through W. Add 2 consecutive rods, W+R; then an additional two, R+G; then G+P; then….

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

(2) Start with W. For each brand-new square, add 2 rods that match the sides of the previous square, and also a brand-new W to fill the edge.