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Two years ago, I bought the book Mathematics for the Nonmathematican, by Morris Kline.

There I learned a new way of solving equations, which is related to the principle that states that any alteration on both sides won't alter the output, Ex:.

$$x+8=4\tag{1}$$

$$x+8-8=4-8\tag{2}$$

$$x=-4\tag{3}$$

In my school, they didn't teach me this way, they taught me that the numbers walk from one side to another and when they do that, they change their sign and in one side we should put all the terms that have x, in the other, we put all terms that don't have $x$'s. Ex:.

$$x+4=2x+7\tag{1}$$

$$x-2x=7-4\tag{2}$$

$$-x=7-4\tag{3}$$

$$-x=3\tag{4}$$

If the $x$ is negative, then you swap the $-$ for $+$ and vice-versa for both terms:

$$x=-3\tag{5}$$

Some days ago I watched a video that stated the importance of teaching long division, the author argued that only long division could teach the concept of convergence and then I got worried with this method, is this way of teaching how to solve equations dangerous somehow? When I learned the first method two years ago, I felt that I could handle my equations better but that could be only wishful thinking.

EDIT: For the ones curious about the video in which the guy says that only long division teaches convergence, you can see it here. Specially this comment.

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Your question reminds me of this. –  user17762 Apr 10 '13 at 3:16
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I don't see why long division is a prerequisite for understanding convergence. But to answer your question, the first way seems better because it reduces to axioms. What does "walking across the equal sign" mean? I think it's just an analogy, perhaps helpful -- I haven't taught kids algebra before. –  user66345 Apr 10 '13 at 3:17
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Your new way is the way my grade 7 teacher taught us. We found the second way easier to understand but I think its because nobody wants to think when they're 12. The first way is more logical but the second way is easier force children to use. :) –  Kyle Schlitt Apr 10 '13 at 3:19
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The first way is undoubtedly the logical way, but I still find myself thinking of it in terms of the second way in practice. I don't think it's too big of a deal either way as long as you understand the underlying concept. –  EuYu Apr 10 '13 at 3:27
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Hrm. At some point when you get experienced with algebra, "move 4 to the other side" and "subtract 4 from both sides" starts to mean the same thing. –  Hurkyl Apr 10 '13 at 3:30

9 Answers 9

The worst thing to teach is that mathematics is a series of recipes to be blindly followed. Mathematics should be about ideas. The reason "walking from one side to the other" works is precisely that it preserves the solutions to the equation. Teaching the mechanics of solving equations without even mentioning that reason should be a criminal offense.

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I suggest much litigation and many lashes for those who teach things differently from my enlightened way of thinking!!! I don't disagree with you... I just find the term "criminal" here rather exaggerated. No psychological or physical harm comes to those who learn the incorrect method here. They just come as disadvantaged in learning mathematics. –  Doug Spoonwood Apr 10 '13 at 4:14
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@DougSpoonwood Don't take it so seriously... –  Pedro Tamaroff Apr 10 '13 at 4:20
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@DougSpoonwood: Hyperbole: A figure of speech in which the expression is an evident exaggeration of the meaning intended to be conveyed, or by which things are represented as much greater or less, better or worse, than they really are; a statement exaggerated fancifully, through excitement, or for effect. [1913 Webster] –  robjohn Apr 10 '13 at 6:06
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The worst example I've ever seen is "Ours is not to reason why, ours is to invert and multiply" oh the irony. –  soandos Apr 10 '13 at 8:09
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@soandos Ours but to do and die? –  Aant Apr 10 '13 at 9:30

The textbook from which I learned basic algebra used exactly that idea. Further, I think numbers walking to the other side teaches use without understanding and should be avoided.

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+1, and I'll add my comment here since this is the most concise answer that relates to it: I also learned the "walk to the other side" version first, and it still causes me problems on occasion - I'll "walk" the 7 in x + 7 = 0 to the other side and forget to change the sign: x = 7 –  Izkata Apr 10 '13 at 13:21

I think the way described in Morris Kline's book is better.

It teaches the basic principle - the meaning of the equal sign =.

In your first example, you add (-8) to both left hand side and right hand side when going from (1) to (2). In other words, you are solving the equation (to find the value of $x$) by keeping the left hand side equal to the right hand side.

I don't see any basic principle behind walk from one side to another.

Convergence is a much more advanced concept than a simple equation. It involves more than the basic principle of equality. If one fails to comprehend what equality means, I don't see how he can understand convergence.

So, yes. It is dangerous to teach students how to solve equations without introducing them the basic principle behind it.

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I agree the second way is not really based on a firm principle but perhaps it may emphasise that the goal is to get a single x term on one side of the equation? –  jk. Apr 10 '13 at 13:46
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@jk. The second way does not teach you much. This question is not about how to solve an equation. My point is that the first way teaches the students better. –  scaaahu Apr 10 '13 at 14:02

I don't see how teaching the "perform the same operation on both sides" process is unusual from a pedagogical standpoint. This was how I was taught to solve equations, and every math textbook I've read teaches it this way.

Remember that when you're solving a equation, the end goal is to isolate the variable you want to solve for such that the variable by itself appears only in and by itself on one side of the equation, and a simplified expression is left on the other side.

The second method simplifies this process for certain cases. However, it does not convey the end goal of isolating the variable. Teaching mathematical procedures without understanding the end goal is only a recipe for failure.

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It is more general (and thus more useful) than walking numbers that change signs, but slightly dangerous because it only holds for certain operations, which the rule you cited fails to mention.

Background

When we solve an equation, we want to know which values for the variables make the equation true. Any derivation step should leave this set of values unchanged (and hopefully make the equation simpler).

Applying the same operation to both sides of an equation will preserve all existing solutions. However, for some operations, it can add new solutions. For instance, if we take the equation

$$ x = x + 1 $$

and multiply both sides with 0, we get

$$ 0x = 0(x + 1) $$

Clearly, x = 4 is a solution to the transformed equation, but not the original one.

More subtly, if we take the equation

$$ x = 4 $$

and multiply both sides with x, we get

$$xx = 4x$$

Now, x = 0 is a solution for the transformed equation, but not the original one.

The correct rule

Applying any invertible operation to both sides of an equation will leave its set of solutions unchanged. An operation is invertible if it has an inverse operation such that: for every value, applying the operation, and then the inverse operation, will yield the original value.

Proof: We have seen that applying an operation to both sides will preserve existing solutions. Any solution to the original equation will be solution to the transformed one. Because the operation is inverse, we can also apply the inverse operation to the transformed equation, yielding the original one. Therefore, any solution of the transformed equation will also be a solution of the original one. Therefore, both equations have the same solutions.

Now, which operations are invertible?

  • Adding any number is invertible (the inverse operation is subtracting that same number).
  • Also, multiplying by any number other than 0 is invertible (the inverse operation is dividing by that number).
  • One might be tempted to think that taking the square root is the inverse operation of squaring a number - and it is, but only if we know the sign of the original number. (9 is the square of 3, but also the square of -3. To undo squaring, we need to know the sign).

A cautionary tale

I fondly remember when our teacher, who was quite overweight, derived:

Let W be my weight, I my ideal weight, and x my surplus weight. We then have:

$$ W = I + x $$ $$ Wx = (I + x)x $$ $$ 0 = x^2 + (I-W)x $$ $$ \left(\frac{I-W}{2}\right)^2 = x^2 + (I-W)x + \left(\frac{I-W}{2}\right)^2 $$ $$ \left(\frac{I-W}{2}\right)^2 = \left(x + \frac{I-W}{2}\right)^2 $$ $$ \frac{I-W}{2} = x + \frac{I-W}{2} $$ $$ 0 = x $$

So you see, I am not overweight at all!

You should now be able to spot where he cheated.

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Why would one need to multiply both sides by $x$ in $x = 4$? I don't get. –  Vÿska Apr 18 '13 at 7:33
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I don't claim one needs to do that. I claim that the rule permits it, and this changes the set of solutions. I chose a trivial example to make that change obvious, but there are other examples where it is less obvious and can result in incorrect conclusions - such as the "cautionary tale" my post ends with. –  meriton Apr 18 '13 at 19:53

The first method includes, in some sense, the second method. You can "walk from one side to another", because any alteration on both sides won't alter the output. For your example

 x+4=2x+7 implies x-2x=7-4, because given x+4=2x+7 we have
 x+4-4=2x+7-4 which implies
 x+0=2x+7-4
 x=2x+7-4 which implies
 x-2x=2x-2x+7-4
 x-2x=0+7-4
 x-2x=7-4

The second method only applies in certain situations also. If I've understood things correctly here, you can only use the second method when you have a group structure. However, you can always alter both sides equally without altering the output, or equivalently alter both sides in the same exact way. So, the first method comes as more general.

So, yes, teaching the second method exclusively is dangerous. It will kill future algebraists (o. k. that's an exaggeration, and a joke... but the second method doesn't come as general of a method as the first).

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Your "new" way is the way I was taught; you do the same thing to both sides of the equation. It was also explained, through demonstration and a lot of practice, that you can do this to cancel out a term on one side, "bringing it over" to the other side with the opposite sign. The goal of this process is to isolate your variable of interest as being the only thing on one side of the equation with no other instances of it on the other side, thus stating what that variable, in singular form, equals.

In my school, they didn't teach me this way, they taught me that the numbers walk from one side to another and when they do that, they change their sign and in one side we should put all the terms that have x, in the other, we put all terms that don't have x's.

This is... an incomplete way to teach the concept, to say the least. It is a simplification of what's going on behind the scenes that may get the student to the correct answer, but why it works remains a mystery, and that can be dangerous when you try to apply these simple rules in situations where the underlying principle contradicts them.

If I were teaching the concept of solving equations through balanced operations to a group of 5th/6th graders for the very first time, I'd illustrate it with the most direct analogy I can think of; a balance scale. Whatever you do, you have to keep the scale balanced. You can add and remove weights, you can tie helium balloons onto the scale, and whenever you see that you have a balloon and an equal opposing weight on one side, you can remove them; through all of this, that needle has to stay in the center. If it moves to the left or right, you have unbalanced the scale, and so what you end up with won't be equal.

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+1 for mentioning a balance scale. This is the way I learned to think about equations. My teacher brought a balance scale and some balance weights. One balance weight was not labeled and we had to find out its weight. No single weight had the same as the unlabeled one, but he gave us one "configuration" in which the balance scale was balanced (one equation). I intuitively understood what operations I could do and this also helped me to understand inequalities. –  moose Apr 12 '13 at 21:24

As the others have said walking to the other side is a simplification of the basic principle that a change made to one side of the equation has the same effect on the other. Another reason to stay away from walking to the other side is it empowers to correctly handle more complex equations.

For example if you have $4x+8y=16$, someone just thinking of variables as walking may have a harder time correctly isolating the x. Rather than thinking of it as

$$ \frac{4x + 8y}{4} = \frac{16}{4} $$ $$ \frac{4x}{4} + \frac{8y}{4} = 4 $$ $$ x + 2y = 4 $$

Thinking like that also makes it easier to avoid mistakes. I'm sure I've done that with problems involving exponentiation and division.

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I was taught to 'walk numbers from one side to the other' (although I don't think they called it that), but I currently use the principle that an alteration on both sides won't alter the output. Your second example (walking numbers from one side to the other) is to me the same, but with some steps hidden:

$$x+4=2x+7\tag{1}$$

$$x+4-2x=2x-2x+7\tag{1a}$$

$$x+4-4-2x=7-4\tag{1b}$$

$$x-2x=7-4\tag{2}$$

$$-x=7-4\tag{3}$$

$$-x=3\tag{4}$$

$$-x\cdot-1=3\cdot-1\tag{4a}$$

$$x=-3\tag{5}$$

If you know the rules and you can correctly make these hidden steps (change the sign, etc), then there is no problem. Otherwise, I make the steps explicit just to be sure I'm doing it right.

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We called the operation of having terms jump across the equals sign "transposition" back in the day... –  J. M. Apr 11 '13 at 3:00

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