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Is it important to convert subtractions to addition, while simplyfing algebraic expressions (For example converting $3x-2y+7x-7y$ to $3x+(-2y)+7x+(-7y)$)? If yes, why and when?

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My opinion is that it should be exercised and thereafter done mentally (not written down) as part of reading an expression. This then allows students and practitioners to directly follow the order of operations, simplifying at each step, and then in the last step recall and apply only the sign rules for addition. Note that the issue of reading/writing additions and subtractions is identical whether we discuss algebraic expressions or purely arithmetic ones.

For example, say we need to simplify $-10-(-12)+(-7)-4$. First we treat all the juxtaposed signs as multiplying and simplify to get $-10+12-7-4$. Then we can read this as, by definition, additions of positive and negative terms to get $-9$.

Note: (1) this follows the standard order of operations (multiply, then add), (2) the entire process requires only remembering the sign rules for two operations (multiply and addition), (3) at each step the expression gets shorter (notably, removing parentheses in the multiply step), and (4) in practice we don't write down the separate added terms, but should be reading it that way mentally.

When I introduce and need to explain the subtraction, then I'll write something like "$2 - 6 = -4$ because $2 +(-6) = -4$", so as to highlight that the latter is not something we'll customarily write down, but is rather part of the mental reading that we do.

A blog article I wrote about this issue: AngryMath: Automatic Negatives, which motivates the structure of the quiz on "Negative Numbers" at Automatic Algebra.

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  • $\begingroup$ I agree with everything here. However, your answer discusses arithmetic expressions rather than algebraic ones, so does it really answer the question? $\endgroup$ – DavidButlerUofA Nov 29 '15 at 19:27
  • $\begingroup$ @DavidButlerUofA: The issue is equivalent. You can take my example and insert variable terms wherever, but I don't think that's really clarifying anything. The question is really about reading and writing the operations of addition and subtraction. $\endgroup$ – Daniel R. Collins Nov 29 '15 at 19:33
  • $\begingroup$ In that case I think it's worth adding to your answer that the issues are the same as those for plain negative numbers. I know it might seem like this is implicit in your answer, but I think it will make your answer better to be explicit. $\endgroup$ – DavidButlerUofA Nov 29 '15 at 19:40
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    $\begingroup$ Also, there is the issue of rearranging an expression to make it easier to simplify, or combining numbers/terms that aren't next to each other. I think this is where the explicit +(-7x) can help to make the point that the term carries its - sign with it. But as you say, it's something you don't want to do explicitly except when it helps to clarify. $\endgroup$ – DavidButlerUofA Nov 29 '15 at 19:43
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    $\begingroup$ @DavidButlerUofA: I added an explicit note about the equivalence of the issue for both algebraic and arithmetic expressions. Thanks for the suggestion. $\endgroup$ – Daniel R. Collins Nov 29 '15 at 20:01
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Why? It much depends on the background. Here are a couple of scenarios.

You have the philosophy of "new math era" in mind in which $x-y$ should be taught as $x+(-y)$, and $-y$ as $(-1).y$.

You have used the so-called counters model to teach the addition and subtraction of positive and negative numbers. In this case, you might want your students to see $3x-2y+7x-7y$ as $3x+(-2y)+7x+(-7y)$ since then they can think of $3x$ plus $7x$ as counting the number of blue counters (representing positive), and $(-2y)$ plus $(-7y)$ as counting the number of red counters (representing negatives).

Please consider the above scenarios is just scenarios not what I am suggesting as a teaching method.

When? If the point of the question is to simplify an algebraic expression like the one you have given, I guess there would be no must unless you have a pure algebraic algebra lesson in mind. However, if the point of the question is to see a certain structure, then it might be useful to convert subtraction to addition. Here are a couple of scenarios.

consider writing $1-x+x^2-x^3+x^4- ...$ in sigma notation. We need to see it as $1+(-x)+x^2+(-x^3)+x^4+ ...$ and then as $1+(-1) x+ x^2+(-1) x^3+ x^4+ ...$.

Suppose you want to discuss with your students that an expression like $a-b+c$ (whether numeric or algebraic) can be structured in two, though equivalent, ways: $(a-b)+c$ or $a + (-b+c)$. In this case, rewriting the expression as $a+(-b)+c$ would relate the situation to the associative property. And you know, associativity comes into play in one way or the other even when you just want to simplify an expression.

Added. Or you want to show them $x .(y-z)$ is just $x.(y+(-z))$, thus we use the distribution property for subtraction. Or $(x-y)^2$ can be seen as $(x+(-y))^2$, thus we can get the identity for the former from well-known identity for the latter.

The bottom line is your students need to see that they can and sometimes they must convert subtraction to addition though they might rarely need it explicitly for the sake of calculations

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  • $\begingroup$ You might also use sigma notation to write the infinite sum indicated as $\Sigma_{n=0}^{\infty} (-x)^n$. That is, one aims to see it as: $(-x)^0 + (-x)^1 + (-x)^2 + (-x)^3 + (-x)^4 + \cdots$ which is slightly different from what you have written. $\endgroup$ – Benjamin Dickman Dec 2 '15 at 10:59
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    $\begingroup$ @BenjaminDickman To be honest, I've never thought of it like that. Thanks :) $\endgroup$ – Amir Asghari Dec 2 '15 at 11:23
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Understanding that $3-2=3+(-2)$ becomes relevant when learning group theory, where you need to have additive inverse elements.

Off the top of my head I can't remember really needing it before that, although I can imagine it might be important in some ways of explaining operations.

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