Getting To Your Destination

You can be the best driver in the world and still you can be driving for hours on end without ever reaching your destination.

Some students have an interesting reaction when they first overcome some difficulties. For example, some people are used to making mistakes at solving equations, because they don't know the rules or they don't know how to apply them. Meaning, they cancel identical terms out of a fraction without realizing it is an illegal move because those terms are just terms, not factors of the whole numerator and denominator. Or they somehow mess up other algebraic steps of solving an equation. They know they are making mistakes, they just don't know what those mistakes are. They are used to getting the wrong answer because they know they make mistakes then cannot even identify. Then they come to get some tutoring and they start finding out what their mistakes are. They learn the correct way to apply the operations to algebraic expressions. At first they seem unsure. Little by little they gain confidence. And then, when they finally stop making the mistakes they had been making before, something interesting happens. All of a sudden they seem confused. They seem surprised to see they are still not finding the solution to the equation, even now when they are not making mistakes any more! They seem to think: "Where is the solution?" They used to think the reason why they were not getting the right solution before was only because they were doing all those mistakes. So, why are they not finding the solution now? They are not making mistakes any more, so where is the solution? Now they seem to keep on going, correctly applying operation after operation to the equation, without getting anywhere near to a solution. Why is that? Well, this is the metaphor I use in this situation. When you drive a car, you have to learn the correct way to use all the controls, the brakes, the steering wheel, the accelerator and everything. If you don't use them correctly, you will crash the car, or you will break the transmission or you will overheat the engine or cause some other catastrophe like that. Once you learn how to correctly use all the controls you can drive safely. Still, that is not enough for you to get to your destination. It is a necessary condition, but by itself is not enough. You can be the best driver in the world but if you keep aimlessly driving in circles all around town when you have to go from Los Angeles to Las Vegas, you are never going to make it. You need to pay attention not only to the car's controls, but also to the road signals. You may even need a map. That is similar to what happens when you are solving equations. Applying each operation correctly and following the rules well at every step is the equivalent of driving safely and operating the car correctly. Now, actually being able to reduce the equation to its simplest form and finding a solution is the equivalent of getting to your destination. You can be correctly applying operations for hours on end, producing more and more equivalent equations like pop-corn in a microwave, but in order to get the final numerical solution, you have to choose what operations to apply and when to apply them. You have to arrange your algebraic steps so that the equation actually gets simpler and shorter with almost every step. That is another skill you have to master but you couldn't see the need for it while you were crashing the car all the time. It's like when you are climbing a mountain and after getting to the top of a hill, you discover another hill in front of you, which you couldn't see before because its view was obstructed by the previous hill you just conquered.

Solving Equations For a Particular Variable

A very basic principle

An equation has one equal sign.

The equal sign divides the equation into left hand side and right hand side.

The two sides may look totally different from each other as expressions but the equal sign says their numerical value has to be the same.

The fundamental principle of equations says that, when two expressions have the same numerical value, if we apply one operation to both expressions, the resulting expressions after the operation is performed will also be equal in value. They will be equal not to the original expressions, but to each other.

So, if A, B and C are three algebraic expressions, and we have the equation A = B, then all of the following will also be valid equations:

A + C = B + C

A - C = B - C

(A)(C) = (B)(C)

A/C = B/C [provided C is not zero]

A^2 = B^2

Square root of A = Square root of B

This fundamental principle is used over and over to solve equations for specific variables, one step at a time.

For example, in solving for x the equation (3x + 1)/2 = 5y - 4, we can do it like this:

1) Multiply both sides by 2 and we get

3x + 1 = 2(5y - 4)

2) Subtract 1 from both sides and we get

3x = 2(5y - 4) - 1

3) Divide both sides by 3 and we get

x = ( 2(5y - 4) - 1)/3

Now the equation has been solved for x in a series of steps, where each step consists of applying one and the same operation to BOTH sides of the equation.

The fact that the resulting expression for x can be simplified to

x = (10y -9)/3

is not relevant here. I am only illustrating the process we use to isolate x one step at a time by applying the same operation to both sides of the equation.

The following YouTube video from InterAlgebra12 shows several more examples:

Word Problems Are Not That Much of a Mystery

Most of it is just common sense

Many times, my students are surprised by how simple some problems seem when I explain them. They go "Is that all there is to it? Can it be that simple? You didn't use any formula!"
A big part of the difficulty students often have with word problems is they think there is or there should be a special type of formula suited for each particular problem. But, for many word problems out there, that is not the case.
If your first reaction to a word problem is trying to remember a formula that would solve that problem, chances are you are never going to remember such a formula, because you have never seen it, because it is not there in any book, and no teacher teaches it as "the" formula for this problem.
There may well be a formula for that particular problem, but the formula is nowhere to be found on record. Because nobody has bothered to figure it out or to pass it down, because even if they did, the formula would be applicable only to that particular word problem and to no other problem. It would be a very limited, almost useless formula.
So, the first thing you have to do with word problems, is to forget about formulas altogether and just read the problem, over and over and over again, as many times as you need to understand what the problem is talking about, what situation it is describing.
You want to really understand the situation, the process described in the problem. You want to understand it as clearly as you see sunlight. You want to be able to express it in your own words, you want to be able to imagine it, you want to be able to tell a story about it, you want to be able to draw a complete picture of it.
Once you do that, the solution presents itself to you naturally, the numbers practically work themselves out. When you really know what is going on, you know what to do, you know what operations to perform, they make sense.
So, again, it's not how to mechanically make the problem fit into a canned formula, but how to make your very own mind wrap itself around the problem completely, with total abandon, accuracy and precision