This question about FOIL, comments and answers made me think about the two roles of $-$: as a sign and as an operator. This struck me because the title

"Why in the FOIL Method the terms are taken with their signs?"

seems odd to me, because in the example given, $$(x+2)(x-3),$$ there are no signs, just one subtraction operator. To get a sign appear, one first would have to convert to $$(x+2)(x+(\color{red}{-}3))$$ to get the red minus sign.

In Germany, we have "Vorzeichen" (sign, literally "pre sign") and "Rechenzeichen" (operator, literally "calculation sign") and I think a similar distiction can be made in english with sign and operator, but I get the impression that the use is rather sloppy.


When first teaching negative numbers, is a clear distinction between the two roles of $-$ made?

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  • $\begingroup$ Explaining to students that $a -b = a+ (-b) = a + (-1)\cdot b$ helps them to understand subtraction as addition of a negative number (where a negative number (-b) represents operation on a number (b) by -1). So one can translate easily between the "two" roles you speak of. $\endgroup$
    – amWhy
    May 2 '19 at 23:13
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    $\begingroup$ For what it's worth, on at least some calculators there actually is a separate button for $\pm$ versus subtraction. $\endgroup$
    – kcrisman
    May 3 '19 at 3:40

It can definitely be confusing, and a clear distinction should be made when teaching. It's not always made, and students can in fact remain lost for a long time. I've seen students who thought $1 \cdot (-1) = 0$, unsure of the role of $"\cdot"$, $"-"$, and parentheses.

As far as symbol, it's not too confusing if when teaching you use parentheses, e.g. "$(-5)$", and call it "negative five" (rather than "minus five"). When students are ready, explain that it's OK to omit parentheses when there is no confusion.

Here's how Beast Academy approaches it (I am not involved in it, but like the books). First they introduce negative numbers, and it's only after some practice that they get to adding a negative number, and later yet to subtracting:

  1. Explain negative numbers on the number line, explaining that $-4$ is read "negative four", etc.
  2. Comparing negative numbers, e.g. $-15 < -7$
  3. Adding on the number line, e.g. "To add 3, we move 3 units to the right", with example of $5+3$ and then $-5+3$.
  4. But if $-5 + 3$ is $-2$, then it nust be that "3 plus -5 also equals -2", but how does that work on the number line?
  5. To add a negative number, we move left, and write it as $3 + (-5) = -2$.
  6. There is a whole separate section to remind students of subtraction: on the number line, we go left to subtract a number, e.g. with $9 - 5$.
  7. We can use the same method to subtract e.g. $5 - 9$.
  8. Another way to solve $5-9$, we can find the number to add to 9 to get 5. Since $(-4) + 9 = 5$, it must be that $5-9=-4$.

So that's how you can explain why $3 + (-5)$ is the same as $3 - 5$. (Here you can draw the students attention to the choice of the symbol for "subtraction" and for "negatives" -- it's the same symbol, because the two concepts are in fact closely connected.) Then there is a whole other section about subtracting a negative.

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    $\begingroup$ Thanks. No relation, I just have and like the books. $\endgroup$
    – DS.
    May 12 '19 at 16:19

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