A perfect square is a number that can be written as an integer raised to the power of 2. This means that perfect squares can be created by multiplying any integer with itself.

For example, 9 is a perfect square because 3^{2}=9. Because of this relationship, 3^2 is often read as "three squared".

Perfect squares can be modeled visually as actual squares.

Table that cotains dot formation in a square formation for each squared number. 1 squared has one dot, 2 squared has 4 dots, 3 squared has 9 dots, 4 squared has 16 dots, 5 squared has 25 dots.

You can see that each formation of dots forms a square.

\displaystyle 1^2	\displaystyle =	\displaystyle 1\cdot 1=1
\displaystyle 2^2	\displaystyle =	\displaystyle 2\cdot 2=4
\displaystyle 3^2	\displaystyle =	\displaystyle 3\cdot 3=9
\displaystyle 4^2	\displaystyle =	\displaystyle 4\cdot 4=16	and so on.

We can use this same reasoning to justify that 0 is a perfect square too because 0 \cdot 0 = 0.

Examples

Example 1

Evaluate 9^2.

Worked Solution

Create a strategy

Use the fact that squaring a number means multiplying by itself.

Apply the idea

\displaystyle 9^2	\displaystyle =	\displaystyle 9\cdot 9	Multiply the base by itself
	\displaystyle =	\displaystyle 81	Evaluate

Reflect and check

We can also count the number of small squares in a 9\times9 grid, which is the same as evaluating 9^{2} and getting 81.

Example 2

Determine whether each number is a perfect square.

169

200

Worked Solution

Create a strategy

If a number is a perfect square, we can build a square grid with the same length and width that has that number of total squares.

Apply the idea

If we build out a 6 \times 6 grid, we get a total of 36 squares and need to add 1 more square to make 37. This does not give us enough units to create a 7 \times 7 grid so 37 is not a perfect square.

6x6 grid with 1 extra square at the lower right. 6x6 squares are labeled.

If we build out a 13 \times 13 grid, we can create a grid that is a perfect square with equal length and width. There are a total of 169 squares in this grid, so 13 \cdot 13 = 169, and 169 is a perfect square.

If we build out a 14 \times 14 grid, we get 196 squares so we need to add 4 more squares to get to 200. This is not enough units to create a 15 \times 15 grid so 200 is not a perfect square.

14x14 square grid with 4 extra squares at the lower right. 14x14 squares are labeled.

Reflect and check

Building out grids can be helpful ways to determine and justify whether a number is a perfect square. It is important to consider whether we have enough leftover units to build a larger grid, so it's always a good idea to try building a grid with a 1 unit increase in length and width.

Idea summary

Perfect squares are numbers raised to the power of two or can be obtained by multiplying an integer by itself.

Outcomes

6.NS.3

The student will recognize and represent patterns with whole number exponents and perfect squares.

6.NS.3b

Recognize and represent patterns of perfect squares not to exceed 20^(2), by using concrete and pictorial models.

6.NS.3c

Justify if a number between 0 and 400 is a perfect square through modeling or mathematical reasoning.

3.04 Patterns with perfect squares

Ideas

Patterns with perfect squares

Exploration