Informally: once you multiply an essence (a “whole” number, positive, an adverse or zero) times itself, the resulting product is dubbed 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, space all square numbers.

You are watching: What is the next number in the sequence below 1 4 9 16 25 36

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

Mathematical background

Objects i ordered it in a square array

The surname “square number” originates from the truth that these certain numbers that objects can be arranged to to fill a perfect square.

Children have the right to experiment through pennies (or square tiles) to view what number of them deserve to be arranged in a perfectly 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 can not be arranged the way. Numbers (of objects) that can be arranged into a square selection are referred to as “square numbers.

Square arrays must be full if we are to counting the number together a square number. Here, 12 pennies room arranged in a square, but not a full square array, so 12 is not a square number.


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The number 12 is no a square number.


Children might enjoy exploring what number of pennies can be arranged into an open up square prefer this. They space not referred to as “square numbers” yet do follow an amazing pattern.

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


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Here room two boards, 3 × 3 and 5 × 5. How plenty of red tiles in each? Black? Yellow?Are any of those square numbers?What if you tile a 4 × 4 or 6 × 6 board the exact same way?Can girlfriend 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 appear along the diagonal of a conventional multiplication table.


Connections with triangular numbers

If you count the green triangles in every of this designs, the succession of numbers you view 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 space in the “spaces” between the environment-friendly ones, the sequence of numbers starts through 0 (because the very first design has no gaps) and then continues: 1, 3, 6, 10, 15, …, again triangular numbers!

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

A connection between square and triangular numbers, seen one more way

Build a stair-step 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 overlook the stepwise edge). Put the two consecutive triangle 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 the the size of the longest pole (as measured v white rods).

Here’s an additional example:

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

Stair measures from square numbers

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Stair steps that walk up and also then earlier down again, favor this, likewise contain a square variety of tiles. Once the tiles are checkerboarded, together they are here, an addition sentence that explains 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 also 1) to build stair-step patterns and write number sentences that define these trends is a nice means to provide them exercise with descriptive number sentences and also becoming “friends” with square numbers.

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Here room two examples. Color is used right here to help you check out what is gift described. Kids enjoy color, but don’t need it, and also can often see creative ways of explicate stair-step patterns that lock have developed with single-color tiles. Or they might shade on 1″ graph document 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 make from pennies can additionally be described by the 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 number sentence.

From one square number to the next: two pictures with Cuisenaire rods

(1) begin with W. Include two consecutive rods, W+R; then another two, R+G; then G+P; then….

See more: Explain Why Do Solid Expand On Heating, Thermal Expansion Of Solids, Liquids And Gases

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

(2) start with W. For each new square, include two rods that match the political parties of the previous square, and a new W to fill the corner.