How To Make Working With Squares Fun in Math?

In mathematics, squares are typically defined as numbers multiplied by themselves. For example, 3*3 = 3². While this is a graspable concept for most people, things start moving fast when we reach x² and quadratic functions. Most people lose their interest at this point.

However, there is a more intuitive definition of a square in the field of geometry. Yes, I mean the classical square shape. In this article, I aim to show that algebraic operations that appear boring and challenging with squares can be understood intuitively by using visual methods. I’ll start with a fundamental outlook and proceed to demonstrate a couple of more advanced results. This should provide you with enough working tools to explore further on your own.

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What are Squares?

Let us start with a single square that has a generic side length of 1 unit (could be centimetre or millimetre or anything else for our purposes). Let us call this a unit square from hereon. It comprises a square by itself and is one unit long. In mathematical terms, it represents the square of the integer 1.

Now ask yourself the following question:

“What is the least number of unit-squares we need to stack together to form a bigger square?”

The answer is that 4 unit squares are stacked to form a 2×2 square. We call this the square of integer 2 (or 2-squared).

Similarly, the next biggest square is 3-squared made of 9 unit-squares. Following that, the next biggest square is 4-squared made of 16 unit-squares. We can apply the same logic to visualise the squares of higher integers.

Now that we have a grasp on how to approach integer squares geometrically, let us jump to more challenging applications.

Difference of Squares

Let us say that there exist 2 squares: a bigger square that is ‘a’ units long and a smaller square that is ‘b’ units long.

Now suppose that we are interested in calculating the difference in their areas. Such scenarios occur often in real life. As examples, imagine situations where you are laying tiles or fitting sections of wooden furniture. One way we could calculate the difference in areas is by placing the smaller square over the bigger square and then calculating the area of the shaded section shown below.

But there is an easier way to do this. All we need to do is cut off a rectangle whose height is the length of the smaller square as shown below. Then, we move it to the side of the remaining portion of the bigger square. The result is a bigger rectangle with a length of (a+b) units and a height of (a-b) units. If we find the area of this rectangle, we get the difference of the squares.

By doing this manipulation, we have reduced the number of multiplication operations from two to one. In plain words, we have made the problem easier for us to solve. The algebraic equivalent of this identity is as follows:

I’ll demonstrate one more application of the visual/geometric approach.

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Proving that any odd number is a difference of successive squares

One of the elementary theorems in number theory states that any odd number can be expressed as a difference of successive squares. Before we look at the exact equation, I’ll demonstrate the entire concept visually first.

Suppose that we choose an odd number, say, 5. We could first represent 5 using five unit-squares in a straight line as below. Then, we could bend these squares to form an inverted L-shape as shown:

Now, let us fill in the empty slots on the lower left with unit squares as well. In this case, it turns out to be 4 new squares.

If you observe closely, there are two squares here, a bigger 3×3 square, and a smaller 2×2 square. Let’s subtract their areas and see what comes out: 3² — 2² = 9–4 = 5. This is an interesting result. We get the same number of unit squares we started with (5).

You can play around with bigger numbers, and eventually, you will arrive at the following conclusion. Any odd number K can be represented as k = (n+1)² — n², where k = (2n + 1). The algebraic version of the same is as follows:

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Scaling Higher Dimensions

I just showed you a couple of illustrated examples of how you can use visual methods to solve problems and intuitively understand mathematics. Consider this a starter pack. The more creativity and imagination you employ, the more fun and ease you will have when solving problems.

To conclude, I’ll share a teaser about how we could scale this approach into higher dimensions. Remember how we visually calculated the difference in areas of two squares?

It wouldn’t take much more mental effort to scale it up to the difference in volumes in cubes. What applications could such a process find? And what about the next dimension? What exists beyond a cube? The fourth dimensional equivalent of a cube is called a tesseract.

You can see a projected rotation (along two planes) of a tesseract above. I’ll let you ponder upon its potential applications, and how we could scale our visual method to play in higher dimensions.

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Further reading that might interest you: How Imagination Helps You Get Good At Mental Math? and Is It Time For Us To Reimagine Regular Education?