Math Tutorials. The Why not just The How

[Newtons Apple]

Hey everyone. I've been facing a problem in my math studies that I can no longer ignore. I'm in college, and I'm taking a college level Algebra class, with the looming Precalc next semester. My problem with math is this... I don't know why we're doing problems. More specifically, I don't know why we follow certain steps. In English, I can reason why and "hear" what makes a proper sentence. The syntax of English and writing comes more naturally to me. With math, and equations, and numbers, the 'why' seems lost. All the tutorials, I've seen will berate you with how to solve equations... Thats just fine, but I want to look at a problem, and be able to logically know what to do.

For example, a problem like

-8x > 10 +3x

I know that I'm supposed to subtract 3x from both sides. But why? No one has ever explained why we do things like this. All my professors just say "Because that's the way its done" but I know, that mathematics has a purpose, but I'd like a reason for it, I think if I can start thinking of problems and equations logically, I can understand and solve them easier... Does anyone know of any books, or websites, or online tutorials, that provides explanations, and reasoning behind Mathematics? Answering Why we follow steps to solving equations, not just How?

 

[gb7nash]

To answer your mathematical question, sometimes it's useful to add a number on both sides of an equation. For instance:

x + 5 = 10

If we have this equation and want to solve for x, it's useful to add/subtract a number on both sides of the equation. The reason why this is so powerful is that we're not changing the equation at all. We're assuming that the left side is equal to the right side, so adding the same number on both sides doesn't change anything. Writing:

x + 5 = 10 is the same as:

x + 5 - 1 = 10 - 1

and:

x + 5 - 5 = 10 - 5

This is the more useful application, since x + 5 - 5 = 10 - 5 is equivalent to x = 5, which is the answer for x.

 

[Newtons Apple]

Well I definitely thank you for the explanation for this specific question, but I just used it as an example. I'm looking more at the broad picture, and wish to get an explanation like yours but for all of Algebra, and precalculus. The reason behind why we do things just seems lost too me... I just can't think logically to see past the numbers. We had to do absolute value equations today and it boggled my mind as to why there's a negative and positive version of the numbers...things like that.. any help?

 

[verty]

Well math is the numbers, so why do you want to think past the numbers? I think you have to provide that part yourself, to find how you like to think of the math.

For -8x > 10 + 3x, there are rules about equivalent statements and that statement is equivalent to one of the form a < x < b or whatever. You have to find that equivalent statement. A theorem of the real numbers says that, if a < b, then a + c < b + c, and so on. Adding on both sides will preserve the equivalence, so it is safe to do so, and it simplifies the equation, it brings it closer to that ideal form.

I think of it as a forest of statements, with paths cut through the forest to isolate patches of trees that are equivalent. But you must find your own rationale/explanation.

 

[symbolipoint]

Well I definitely thank you for the explanation for this specific question, but I just used it as an example. I'm looking more at the broad picture, and wish to get an explanation like yours but for all of Algebra, and precalculus. The reason behind why we do things just seems lost too me... I just can't think logically to see past the numbers. We had to do absolute value equations today and it boggled my mind as to why there's a negative and positive version of the numbers...things like that.. any help?

With that posting and your original post on this topic, you give the impression of being in a course beyond your level. Did you study "Elementary Algebra" and then "Intermediate Algebra"? If you did those, you would be far more secure in understanding and using properties of numbers, basically for Arithmetic (since beginning and intermediate Algebra are both mostly generalized Arithmetic).

 

[SteveL27]

Hey everyone. I've been facing a problem in my math studies that I can no longer ignore. I'm in college, and I'm taking a college level Algebra class, with the looming Precalc next semester. My problem with math is this... I don't know why we're doing problems. More specifically, I don't know why we follow certain steps. In English, I can reason why and "hear" what makes a proper sentence. The syntax of English and writing comes more naturally to me. With math, and equations, and numbers, the 'why' seems lost. All the tutorials, I've seen will berate you with how to solve equations... Thats just fine, but I want to look at a problem, and be able to logically know what to do.

For example, a problem like

-8x > 10 +3x

I know that I'm supposed to subtract 3x from both sides. But why? No one has ever explained why we do things like this.

There are two general principles involved here:

1) You can (and must) do the same thing to both sides (as long as you don't multiply or divide by a negative number, in which case you have to reverse the inequality sign);

2) You want to isolate the x-term entirely on one side.

So from (2) you see you want to get that 3x over to the left. How could we get rid of 3x on the right hand side? By subtracting 3x from the right side. But (1) says that we have to do the same thing to both sides. So you subtract 3x from both sides to get

-11x > 10.

Does that help?

 

[Newtons Apple]

Again I appreciate the input Steve but I already knew how to do it, what I was looking for is why we must keep both sides balanced? I mean obviously I know if you change one side of an equation then it doesn't make sense anymore, but even past that, I guess what I'm fundamentally asking for guys is, do any of you know if a math book or site that really explains why we follow steps in Algebra?

 

[symbolipoint]

Again I appreciate the input Steve but I already knew how to do it, what I was looking for is why we must keep both sides balanced? I mean obviously I know if you change one side of an equation then it doesn't make sense anymore, but even past that, I guess what I'm fundamentally asking for guys is, do any of you know if a math book or site that really explains why we follow steps in Algebra?

Why keep both sides balanced? Because the step must still give an equation. If you change one side of the equation but do not change the other side in the same way, then you created an inequality.

 

[dalcde]

The objective to subtract 3x from both sides is to simplify the statement so that it is easier to work with without changing it meaning.

 

[hotvette]

Again I appreciate the input Steve but I already knew how to do it, what I was looking for is why we must keep both sides balanced? I mean obviously I know if you change one side of an equation then it doesn't make sense anymore, but even past that, I guess what I'm fundamentally asking for guys is, do any of you know if a math book or site that really explains why we follow steps in Algebra?

Rules of mathematics are needed to correctly achieve an objective. In the simple case of x - 5 = 7, the objective is to find out what x is. To achieve that, one must follow the rules (i.e. natural truths) of algebra. In calculus, you can't mathematically find the area under a curve unless you follow the rules of integration. In language, the objective of communication is achieved if everyone follows the rules of grammar. Rules and procedures are everywhere. You can't drive your car to the store (objective) unless you follow the rules of 1) starting the car engine, 2) pressing on the accelerator, and 3) navigating the roads to the store, in that specific order. The hard part is that the objective of mathematics can often seem abstract and meaningless. Nobody will question the objective of verbal communication or going to the store but they might wonder why you need to solve the equation x - 5 = 7 or find the time it takes for a rock to hit the ground if it is thrown straight up at a certain speed.

If you are looking for a list of rules (of alegebra) and a description of the purpose of each one and consequence of not following it, I think you'd be hard pressed to find it. I think language provides a good parallel. I think it would be tough to precisely describe the reason to use the correct preposition or verb tense other than a general statement that you'll likely be misunderstood if you don't. In the case of math, there might be some cases where a precise reason and/or consequence might be possible, but for the most part it's the general reason that 'you will not achieve the objective'. At least that's true for basic topics like algebra. You kind of just have to accept on blind faith that following the rules will produce the desired result. That's my view, anyway.

 

[chiro]

Hey Newtons Apple and welcome to the forums.

The idea of all these "rules" is that if you are given a constraint (a constraint is just some mathematical rule) of some sort (like x - 5 = 10, or x + 3 < 10), then basically you can't change the definition of the constraint, but you can use these "rules" to transform it in a way that doesn't change it's meaning.

All of the rules of changing equations and inequalities are based on doing this. If the meaning of the constraint is unchanged after some operation, then the operation you just did is ok.

If you do further maths, you'll find that there will be problems where you have some constraint (like an equality or some inequality) and you will look instead for an approximation, and when this happens you'll do things that you can't do when you want to preserve an equality or inequality which might confuse you when your first see it, but remember that approximations are different from equalities and inequalities.

 

[paulfr]

Think of Mathematical equations [ algebraic, logarithmic, exponential, trigonometric, etc. ] the way Richard Feynman did.
It is a kind of game. And the game is to find the value(s) of x that make the equation true. That is when the left side is the same as the right side.

Now every time you do something to the equation you end up with what we call an "equivalent equation". And at some point the equivalent equation is of the form" x = something". This is the solution. Think of it like this. You have a hollow sphere [like say a basketball] and want to know what is inside. But there is only one tiny window somewhere that let's you see inside. So you keep rotating and turning the sphere until you can see what is there. Finding equivalent equations is like viewing the same thing from different points of view.

For an equation with one linear variable [power = 1] there is only one number that satisfies this objective. But if the power of the variable is ever 2 then there are 2 numbers/roots/solutions that make the equation true. This is the Fundamental Rule of Algebra. An equation of degree "n" has n solutions.

You can try to find this number by trial and error if you like.
Keep putting numbers in for x and see if the two sides end up equal to each other.
But it is much more effective to follow some rules that get you to a solution.

Now the three rules of equations are [my rules for my classes]
1/ What you do to the left side you must do to the right side
2/ Isolate the unknown variable to one side to get to ... x=
3/ Invoke the Inverse Function whenever possible.

Inverse Functions ... that is for another thread.
Or Google or Wiki to learn about them.

 

[operations]

Let me try to help. We only add to both sides because it is logical. Let's forget a little bit about math. If you say:

red = red

That is a true statement. But if you say:

red + yellow = red

That is a false statement. To make the above statement remain true, you have to say:

red + yellow = red + yellow

You have "add" yellow to both sides, otherwise you are just saying nonsenses... This have absolutely nothing to do with mathematics. You just add yellow to both sides so people won't think you are crazy saying that red equals orange :)

 

[Newtons Apple]

Thanks for the replies everyone, I guess after sort of shying away from math for so many years its going to take a while to be able to competently solve equations. What bugs me is all the different nuiances and rules of algebra. It just seems like there's a special way to do every single problem in algebra, and once I remember and think about one set of rules, I forget another set of rules. For example, all the different methods to factor equations. To me, a person who isn't steeped in math, it looks like a bunch of random methods. How do people look at an expression and sort of surmise how to do it and what steps to take? Its just so frustrating seeing people who just 'get it' and others like me, who just don't have that knack.

 

[symbolipoint]

Thanks for the replies everyone, I guess after sort of shying away from math for so many years its going to take a while to be able to competently solve equations. What bugs me is all the different nuiances and rules of algebra. It just seems like there's a special way to do every single problem in algebra, and once I remember and think about one set of rules, I forget another set of rules. For example, all the different methods to factor equations. To me, a person who isn't steeped in math, it looks like a bunch of random methods. How do people look at an expression and sort of surmise how to do it and what steps to take? Its just so frustrating seeing people who just 'get it' and others like me, who just don't have that knack.

You described how some of us see equations and inequalities at whatever level we have developed comfort: we see the relation and we know what steps to do to either factor, or to solve for an unknown. The level at which this is easy or difficult depends just on how well each of us has developed.

We develop our comfort based mostly on intense study, and many many practice exercises. Some of us really study and practice some courses and exercises more than once. The more you study, for example, "Algebra 1", the better you become with it. Studying beyond a course also helps, for example, studying "Algebra 2" will rely on everything you were supposed to learn in "Algebra 1". Also, the exercises are a bit more involved, and you would do many of them. If some day you study Algebra 1 all over again, maybe much of it will be easier than it was the first time. You get better at whatever you study with more practice and continued detailed study.

With Elementary and Intermediate Algebra, you are mostly focusing on properties of Real Numbers. Those are most of the rules. After enough study and practice, you just know them, as if they were part of your language. When you deal with equations and inequalities, you usually do NOT see more than two steps ahead, and that is perfectly acceptable. You work one step, and then you decide (or sometimes just SEE) what step to do next; sometimes you can combine steps (like two at once, as long as you are applying the properties correctly).

Knowing how to study is also very important. How do you, yourself study Algebra? What exactly is your process? One reason certain people may feel the way you described (forget the rules, all the different methods, frustrated that others just "get it") may be the result of just how you yourself study. That's why I asked.

 

[paulfr]

Google for the Algebra Cheat Sheet
Look for the one from "tutorial.math.lamar.edu"
There you will find everything you need to know to become
more comfortable with all sorts of problems in Math.

Summary sheets like this one are an outstanding study aide
for remembering what you learn. No one goes back to read
the book 5 times. But a study sheet can be read easily every day
until it is like vocabulary; you just know it from repetition and use.

 

[206PiruBlood]

I think it should also be noted that these introductory courses generally focus mostly, if not entirely on applied aspects. We have the benefit of knowing which topics are widely applicable and seem to focus on these without mentioning just how broad the field really is. At the heart of mathematics is deductive reasoning - beyond that, the possibilities are essentially unlimited. One thing mathematicians do is create consistent, independent rules and see what consequences they can deduce from these rules. The trick is to make rules from which you can deduce useful relationships. One such system of rules is the properties of real numbers and after deducing a number of simple theorems you can solve some very useful problems: such as your inequality which could represent many physical situations.

I think your situation is pretty much the norm. One usually takes a course in logic/set theory or some other similar one before delving into the more theoretical topics; and its here they focus more heavily on motivation and reasoning.

I think a great book for you would be "What is Mathematics" by Richard Courant. https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20

I think I started each line with "I think"