Simplifying Math and the 4×2 Method

-Changing our mentality when solving equations

Posted on July 4th, written by Omar Said

When I was young, my parents insisted that I start learning as much math as I could as early as I could. This led to me spending many summers attempting to solve equations that, for all intents and purposes, I wasn’t fully ready to solve. Looking back, while it definitely did come with a lot of seemingly needless struggle, I can't deny that I did learn a few strategies that I use while working on university assignments today. These strategies can’t be found in a textbook and aren’t taught in a school and are (more or less) entirely my own. Today, I want to share with you what I believe to be both the easiest and most effective of these strategies. A strategy that made simplifying long stretches of math straightforward and practical. I call this strategy the 4 x 2 method.

The 4 x 2 method is a strategy I often use to simplify equations rooted in multiplying polynomials. I use it because, for all intents and purposes, it allows me to know I'm on the right track without the tedious work of simplifying these equations the traditional way. Let me break it down. We all know that 4 x 2 is equal to 8. We also know that 8÷2 is equal to 4 and that 8÷4 is equal to 2. The crux of this strategy is the simple fact that the relationships between 4, 2, and 8 are so trivial that, whenever we want to multiply or divide using polynomials, we can use the 4, 2, and 8 as stand-ins to both understand the type of answer we are looking for as well as the general way one would go about obtaining this answer. In other words, the 4 x 2 method is simply a half-step between full-on simplification and solving the rote equation. Through this half-step, we both make solving the simplified version of the equation simpler, but we also gain enough information to review our final answer and make sure it makes sense.

Here is a simple example of exactly what I'm referring to:

Now, you might view the answer to the equation above as being so obvious that it barely needs the 4 x 2 method at all. However, what I'm trying to make you see here is how easily the logic I'm presenting within this argument holds true. I'm contending that if 8/2 is equal to 4, then x should be equal to 24/2. My logic holds true because, for all intents and purposes, both of these questions are more or less the same, just with slightly different mechanics. By solving the easy question that barely requires any thought, we have a blueprint for how we solve the harder, more complex questions as well. This is, in essence, exactly why the 4 x 2 method works without requiring one to spend time simplifying equations in the traditional way.

Now, you don't necessarily need to use 4 x 2 for this method, it generally works with any set of numbers whose relationship works in a similar way, for example 3,2 and 6 can be used in the exact same way. In fact, you don't even need to use the method at all if you find it to be cumbersome or a waste of time. The real reason I adopted the 4 x 2 method was to help myself understand questions and put complex equations into terms I could mentally understand. If there is one thing I want you take from this post, its the idea that you can make your own mathematical tools to help you conceptualize questions and that creating these tools both allows you to develop your critical thinking as well as become a better mathematician at the same time.