Question
Dear Mitch,
I'm a 5th grader in elementary. I'm having problems with adding and subtracting negatives and positives. Is 3-(-40)=-43 or -37 i just don't get it. My teacher says that turn the subtracting sign into a adding sign and make the numbers all positive but when your done do you leave the number a positive or negative sign i really need help please respond as soon as you can.
sincerely,
Sara
P.S.My teacher says add the oppisite number
Answer
Sara,
Sometimes when you're stuck, the best thing to do is just forget about everything you sort-of-almost-kind-of-know about how to do the thing you're trying to do, and start over at the very beginning.
Why? Because maybe when you wrote down something your teacher said you left out a word, or your teacher left out a word (little accidents like that probably happen a hundred times a day to every busy person, and we don't notice it because most of the time tiny mistakes don't matter much and get fixed along the way without making much of a difference. (Did you ever try to write with the cap still on the pen or start putting your left shoe on your right foot? ... I have, just about every day!!)
BUT IN MATH, those little mistakes can quietly grow if they're not caught in time and lead to BIG frustrations later on...)
So, let's take it from the very beginning. . .
As soon as a person (like you) starts to learn that there are things called negative and positive numbers, it is a good idea to try to change the way your mind reads the little horizontal line that you were always taught meant "minus" (or "take away" or "subtract").
Instead, try to think of (and read) that line (-) as "the opposite of".
So, "-night" = the opposite of night
= day.
So, in MATH:
"-Yes" says 'the opposite of yes', and that means NO.
So "-5" says 'the opposite of 5'.
And if you look at a number line or a thermometer (which is really just a vertical number line (a number line going up and down instead of sided to side), you see that you get the opposite of something by folding the number line along the zero, and then you'll see that every number lands on its opposite.
Well, you can't usually fold a number line because it's a sticker on your desk or a poster stapled to a wall, and you certainly can't fold a thermometer! So you do it in your imagination. If that's hard to do, then try drawing a number line on a piece of paper and folding that one. Tracing paper works best because you can see right through for your answer!
Okay, so now look at this:
- - -5.
Weird?
Well, start at the number and work your way out to the last - on the outside.
So, it says: the opposite of the opposite of the opposite of 5.
First time: the opposite sign next to the 5 lands that 5 on the folded number line at the five BEFORE the zero (or BELOW zero on the thermometer). (That five down there (or back there) is called 'negative five' or 'five below zero'.)
Then the next step you take is with the second opposite sign, and it's folded back to the 5 AFTER or ABOVE zero. So those two moves cancel each other out because you're back to where you started before you looked at any of the little lines. So now you're left with -5, or the opposite of 5.
(If someone asked what's the opposite of the opposite of yes, you'd think, well...going from the inside out, opposite of yes is no, and the opposite of that is yes. So - - yes = yes. And that's true even though - yes is no.
Now, when you change your reading of - from "minus" to "opposite of", you're left with nothing to DO because you lost your verb (minus). BUT you need to do something, and here's what you do: You add them. Why? BECAUSE: If I gave you 2 apples and then 3 apples, when you look at your pile to see what you have, you do something: you add:
2 +3
And you know you have 5 apples. (Adding is done in our head quietly all the time).
So: 3 - (-40) =
Means 3 added to the opposite of the opposite of 40.
WHAT???
LISTEN:
(You can always stick a little plus sign in when working with positives and negatives to help you and it doesn't lead to anything wrong, which is NOT TRUE if you stick in a little minus or 'opposite' sign).
So, 3 - (- 40) is
3 + opposite of opposite of 40) =
3 + 40 (remember: the opposite of the opposite of yes is yes)
And you get 43.
Yes, 3 - (-40) = 43.
Here's a TRICK: Whenever you have a - next to another - like this: - - you just fill in that tiny space between them with a vertical line, so - - becomes +.
So look how easy it can be:
3- (-40)=
3 + 40 =
43.
BUT you can only do this trick for PAIRS of opposite signs next to each other.
SO, - - - 40 = - +40.
And if you do everything right along the way you don't have to change any sign at the end, because your answer will already be the right sign.
For the example you gave me, you do not have to worry about the parenthesis and can ignore them. BUT very often in math you cannot just ignore parentheses. You cannot ignore them when more than one little thing is going on inside them. Then the rule is take care of whatever you're being asked to do INSIDE the parenthesis FIRST, before going back outside to the rest of the equation. And only after you finish doing everything in the parenthesis, and then the thing right next to the parenthesis, can you get rid of them. So: If you have 3- (-40 -5), you go first inside the parenthesis and look at (- 40 - 5).
Remember you can add little plus signs to help you.
-40 - 5 =
- 40 + -5
So it means the opposite of 40 plus the opposite of 5. Usually when we could look at numbers on a number line it helps, but in your head just think: Usually numbers move from zero to the right but with 'opposites' (or 'negatives') we move the opposite way. So it's 40 jumps to the left of the zero, (or below the zero on a thermometer) and (or "plus") another 5 jumps to the left (or downward on the thermometer) and we get the opposite of 45 or -45. So for this:
3 - (-40 - 5) we have
3 - (- 40 + - 5) =
3 - (- 45) =
3 + - (-45) =
3 + opposite of - 45 (opp of opp of yes = yes, so the opp of the opp of 45 = 45.)
3 + 45 = 48. DONE.
P.S. When you say your teacher said turn the subtraction signs into adding signs and make all the numbers positive, your teacher means only for ones where you have pairs of opposite signs. You change into every pair into a plus. Like for this one:
3 - - 40 becomes 3 + 40, yes, because that's what the opp of the opp here means. BUT, USUALLY your teacher (and I) would tell you that when you change a minus into a plus you are changing the thing that comes after it too, and in that case you'd be changing the positive number into a negative number. So you have 7- 4 = 3, or you have 7 + - 4 = 3.
I promise you, this topic is one of the most confusing in all of math, but after you practice and practice and practice thinking about it in different ways, it just happens in your head, like reading probably did when you were a little younger.
Just keep at it!
Hope this helps,
Mitch