Hopelessly lost in algebra land

October 5, 2009 6:40 AM

I am having the hardest time working with positive and negative integers in elementary algebra. Please hope me.

I'm back in college after 20+ years and have gone back in time to elementary algebra. I didn't get it in high school and I'm still having a hard time getting it now. My specific problem is adding and subtracting positive and negative numbers, as well as accurately reading them in an equation. For example:

5(-2) - 4(-2) + 3

this has my eyes crossed. I want to rewrite it as -10 - 8 + 3 which is I'm pretty sure incorrect. I have to get this straight now because otherwise I'll be making sign errors all through the rest of my class, even if my math and rest of the problem solving is correct.

I have made flash cards with the "rules" for addition and subtraction with signed integers and refer to them but sometimes I still come out with the wrong answer.

Does anyone have any other ideas or tips for how to get over this mental block I have with the whole signed integer thing?

posted by hollygoheavy to education (20 answers total)

If I might make a notation suggestion which helped many, many of the kids I tutored in math, you'll want to do two things:

1) Rewrite all negative numbers with a higher, shorter dash. Instead of -, do a ^-.

5(-2) - 4(-2) + 3
5(^-2) - 4(^-2) + 3

2) Next, rewrite all subtraction operations as addition, with the sign of the very first number in the factor changed.

While subtraction and negative numbers are intimately related, I think it was a bit of a mistake to use the exact same symbol. This confuses a lot of people.
posted by adipocere at 6:45 AM on October 5, 2009 [1 favorite]

When you subtract a negative number, it's the same as adding. So that equation should read:

-10 + 8 + 3

which simplifies to:

-10 + 11 = 1


I'm not very good at math myself, and I find that one of the things that trips me up most is rushing through problems. If you can force yourself to go really, really slowly, you should be much less likely to get stuck on a problem.
posted by oinopaponton at 6:46 AM on October 5, 2009

Negative always wins. It's got more life force than a positive. to wit:

a negative times a positive? Negative wins.

A positive times a negative? Negative wins.

A negative times a negative has so much power that it snaps and turns positive. (Think of them like in grammar:
"I'm not doing this" (1 negative)
"I'm not never doing this" (I'm not NEVER doing it, so I'll do it...eventually, so it's an affirmative. Two negatives make a positive.)

So the right way to read that formula is -10 + 8 + 3. (answer = 1)

I just hammered it in my head that any time I saw two negatives together it made positive.

Now if the equation said 5(-2) + 4(-2) + 3, you'd have -10 - 8 + 3 = 15.

Tell me I just did that right? I think you're overthinking the spacing. minus for is the same as negative 4 in this problem. Right? Now you've got me overthinking it.
posted by TomMelee at 6:48 AM on October 5, 2009

Negative always wins. It's got more life force than a positive. to wit:

a negative times a positive? Negative wins.

A positive times a negative? Negative wins.

A negative times a negative has so much power that it snaps and turns positive. (Think of them like in grammar:
"I'm not doing this" (1 negative)
"I'm not never doing this" (I'm not NEVER doing it, so I'll do it...eventually, so it's an affirmative. Two negatives make a positive.)

I remember it as "mixed minus, pure plus"
posted by I_pity_the_fool at 6:50 AM on October 5, 2009

You guys are fantastic!! Those are all very helpful :) Any other mind tricks for math would be great.
posted by hollygoheavy at 6:51 AM on October 5, 2009

Use lots of brackets - more than are strictly required and write out multiple steps.

So you'd first write:
(5(-2)) - (4(-2)) + 3

Then carry out *only* the innermost calculation in each set of brackets:
(-10) - (-8) + 3

Replace all double minuses with plusses (less confusing):

-10 + 8 + 3


Obviously this takes way longer and can be cumbersome, but this is what I used all through university to prevent sign errors and other algebra mistakes.
posted by atrazine at 6:55 AM on October 5, 2009 [1 favorite]

Work through each of the pieces in very small steps. So you've got

5(-2) - 4(-2) + 3

First do the multiplication. Completely ignore addition and subtraction, and don't try to combine signs! You can deal with that later. Maybe write it out with a lot of space in there. It then becomes:

5(-2)      -      4(-2)      +      3

(-10)      -      (-8)        +      3

(-10) - (-8) + 3

*Then* you can combine signs. Subtracting the negative makes a positive, so it turns into

-10 + 8 + 3

And there you go!
posted by soma lkzx at 6:58 AM on October 5, 2009

Imagine (or draw) a number line on the ground, and a little guy standing on the line (I used to do this at the front of my class). Positive numbers to the right, negative to the left. Always start at zero, facing to the right (positive). If you add two numbers, 1+2, he goes one step right, then two steps right. He would then be at the answer, 3. For negative numbers, he walks backwards; -1 + -2 means he steps back one, then steps back two more. He would then be at negative three. Add two and a negative five (i.e. 2+(-5)) and he steps two forward, and fave back, winding up at negative three.

Now say you want to subtract. 5-2. Again, start at zero. Take five steps tot he right. Now, subtraction does something a little different - you turn around. So after your little guy takes five steps to the right, he pirouettes, and now faces left. Then he takes two steps forward, which is to your left, and ends up at three.

Here comes the kicker. Say you have 2-(-3). He starts on zero, facing right. Takes two steps and is on positive two, as expected. Now it's subtract, so he pirouettes to face left. But it's subtract a *negative* three, so he walks backward three steps, winding up at positive five. You can see, then that he could just as easily have just taken three steps forward before he turned around, and wound up at the same place. I.e., 2-(-3) is the same as 2+3.

For your example, there's multiplication so it's a little more complicated. Mulitplication is like asying, "do this, that many times." So you could say, "Take two cards from this deck" five times, and the person to which you speak would have taken ten cards by the end of the exercise. 5(-2) means, take two steps backwards, five times...which gets your little guy to negative ten.

Then he turns around so he's facing left (because of the subtraction symbol) and takes two steps back. And again. And again. And again (four times total). Equivalently, instead of turning around and walking bakwards, he could have taken eight steps forward, before he spun around. At any rate he'd be at negative two.

Then there's a third step. For each step, of course, he starts each step facing right. So he's at negative two, facing right, and takes three steps. He is now on *positive* one.
posted by notsnot at 7:10 AM on October 5, 2009 [2 favorites]

Think of numbers as a direction and a distance from a point (zero) - say left and right along a line. Add x means "go right x steps". Subtract x means "go left x steps". Multiply by x means "Make your distance from the zero x times bigger". Multiply by minus 1 (or a number with a minus sign) means "rotate 180 degrees about zero". So -2 x -1 means "make a line that goes 2 steps left, then rotate it about zero" - which is the same as "go 2 steps right", i.e. +2.

So

5(-2) - go 2 steps left, then 5 times the distance - is 10 steps left, or -10

- 4(-2) - go 2 steps left, then 4 times the distance (making 8 steps left), then rotate it about zero - making 8 steps right, or +8

+ 3 - means 3 steps right.

So now we add them together. 10 steps left, 8 steps right, 3 more steps right, leaves us one step to the right of where we started: i.e. +1. If it helps, draw it out on a bit of paper.

(As a bonus, this picture can make things easier when you come to deal with complex numbers...but I won't confuse the issue here by going into that)
posted by Electric Dragon at 7:19 AM on October 5, 2009

Another thing I've found people have trouble with is the lack of multiplication signs. Two expressions next to each other are being multiplied, unless some sign intervenes. So, you need to wrap the sign in parens otherwise: 3(-2) is multiplication but (3-2) is subtraction. It helps to insert the implied multiplication signs.
posted by Obscure Reference at 7:28 AM on October 5, 2009

For this kind of thing it helps to think like a computer. When you tell a computer to calculate, for instance five times negative two plus negative three times five squared, it needs break down the problem into tiny steps and do them in the right order to get the answer. In algebra, the order of operations is Exponents/Roots, then Multiplication/Division, then Addition/Subtraction.

5 * -2 + -3 * 5² : Exponents go first, so start with 5².
5 * -2 + -3 * 25 : Now multiplication, so both sides.
-10 + (-75) : Adding two negative numbers is just like normal addition.
-85

If you're getting confused about what order to do things in, just go through the problem before you start and add parentheses in the order that you'll be doing the operations in. So for the problem about you would have:

(5 * -2) + (-3 * (5²))

Then start in the innermost parentheses and work your way out.
posted by burnmp3s at 7:37 AM on October 5, 2009

While subtraction and negative numbers are intimately related, I think it was a bit of a mistake to use the exact same symbol. This confuses a lot of people.?

huh? I don't see in what sense they are different.
posted by mary8nne at 8:12 AM on October 5, 2009

huh? I don't see in what sense they are different.

The expression "-x - y" has two minus signs that mean two different things. The first minus sign represents a unary negation operation that basically means "take the value of x and flip the sign." The second minus sign represents binary subtraction operation that means "take the value of y and subtract it from the value of -x." It would be less confusing if the negation operation used a different symbol, but if you know the rules you can figure out what any given minus sign means in context.
posted by burnmp3s at 8:30 AM on October 5, 2009

I am sort of a visual learner. So for me when I have -10 - -8 the two - before the 8 combine to form a +

Make sense?
posted by nestor_makhno at 9:20 AM on October 5, 2009

I look at it this way:

Like adipocere said ("rewrite all subtraction operations as addition, with the sign of the very first number in the factor changed"), when you subtract one number from another you can also think of it as adding the negative of that number to the other number. For example:

100 - 50 is really (100) + (-50).

Therefore you can break 5(-2) - 4(-2) + 3 down into:

(5)(-2) + (-4)(-2) + (3)

Do all your multiplying:

(-10) + (8) + (3)

Add all the numbers that are the same signs. 10 doesn't get added to anything because it's the only negative number. 8 and 3 get added because they are both positive numbers:

(-10) + (11)

And, for days when you really cannot do math (I have those days), flip things around to look right:

11 - 10

Voila! That's easy! 11 - 10 is 1!
posted by halonine at 9:26 AM on October 5, 2009

(Also what nestor_makhno said.)
posted by halonine at 9:27 AM on October 5, 2009

When I was a kid my brother taught me to think of plus signs as the good guys and minus signs as the bad guys. Two bad guys would obviously be friends.
Also, maybe it would help to think that "-2" is really "0 - 2"?
posted by lucidium at 9:47 AM on October 5, 2009

TomMelee: Now if the equation said 5(-2) + 4(-2) + 3, you'd have -10 - 8 + 3 = 15.

Just so you're not getting more confused, that would actually be -15.
posted by teg at 10:27 AM on October 5, 2009 [1 favorite]

my algebra teacher told us the following for keeping negatives straight and it has still stuck with me...

think of numbers like people, negative numbers are bad people, positive numbers are good people.

when something bad happens to a good person the result is bad
negative x positive = negative

when something good happens to a bad person the result is bad
positive x negative = negative

when something bad happens to a bad person the result is good
negative x negative = positive

when something good happens to a good person the result is good
positive x positive = positive

As far as subtracting goes, I always had "Two wrongs DO make a right!" which meant that subtracting a negative number (two wrongs) you actually add (right).
posted by magnetsphere at 4:12 PM on October 5, 2009

I'm late to the party, but I teach this stuff and let me add one thing that also helps students:

Once you get to -10 + 8 + 3 think in terms of money:

if you owe 10 dollars (it's negative) and I give you 8 dollars (it's positive), you would still be behind by 2 dollars; - 2.

If I then give you three dollars, you would now be ahead by one dollar. Answer: 1

This seems to really help students in problems like -5 + ( - 3) = - 8.

If you owe one person 5 dollars, and someone else 3 dollars, you owe a total of 8 dollars.

Hope that helps!
posted by wittgenstein at 6:39 PM on October 5, 2009