I am tutoring a 9th grade boy in Algebra I. The greatest stumbling block is dealing with the issue of a subtraction sign vs. a negative sign. Also, he has trouble “adding” positive and negative numbers together.

I have shown him examples on a numberline. I have tried to explain with money (like he makes $15, but owes his dad $20). Ultimately, how do I explain that 2 - 5 REALLY MEANS 2 + (-5)?

Or, consider (3x-4) - 10(x-7). He knows to distribute the 10, but he is hesitant to distribute the “-” along with the 10! I think it leaves him unsure what sign should then appear between the two parenthetical terms. HE’s seeing the “-” a minus, a fixed symbol linking the two parenthetical expressions. But, I want him to think of it as a negative sign attached to the 10, so both the 10 and the “-” get distributed together!

Ultimately, how do you explain that +(-10x+7) means to ADD a negative 10x term and add 7? How do you teach someone that adding sometimes means to subtract, and subtracting a negative from a negative is like adding two negatives? What other concrete examples can I give him to make it clearer?

The primary problem is the loose nature of the language! This is compounded the need to unteach years and years of thinking subtraction is always “take away”, and a negative number is a different animal than subtraction!

How can I teach it so he can so he it for himself???
I know I had the very same confusion he has, but I don’t know what made the light dawn.

Try creating (before hand) a number of problems like this: - 10(x-7), and ask him to simplify them. He does one (presumably incorrectly). You show him how to do it properly then remove the piece of paper you did it on and have him do it again. If he gets it wrong, show him again and then have him try again. Continue until he gets it right. If he gets it right, have him move on to the next one.

Give him a big ol’ equation (IE written in size 72 font). 1–1 and 1±1. One’s shorter’n the other, and they mean different things. A minus sign simply means that the number is to be subtracted from another number. A negative sign indicates a number’s placement on (yup, back to this) the number line.

You might also remind him that absence of a negative sign really means a number is positive, and draw up a few equations like this:

+2 +5=7
-4 +3=7

etc

I suppose showing him how a double negative becomes a positive might be too confusing, yes? No?

Is he familiar with the practical applications of the commutative (I think that’s the one) law of mathematics? Given some instruction/reminder of that, if he’s familiar with it he should be well off:)

If this is part of your original example, you need not “+(-10x+7)” but “+(-10x+70)”. Or, less confusingly (to me, anyway), work it backward. Start with -7X+66. Then go from there to 3x–4–10x+70. Then go to (3x–4) –10(x–7), or (3x + -4) + -10(x+ -7) if that works better for him.

Does he know what the words mean in the context of the language? EG that adding is to take the sum of the two values, and subtracting is to take the difference?

Ooh. Maybe this’ll help. Know the old saying “when two vowels go walking, the first one does the talking and says its own name”? Well, adding a negative number is the exact opposite. When the plus sign and minus sign go walking, the minus sign does the … action, in this case.

Ask him for examples of what he has learned that have given him his definitions of mathematical actions. From that, you should be able to extrapolate a way to get from where he is to where he ought to be/what the truth is. You say he believes that subtraction is always taking away. Subtraction is nothing more than the adding of a negative number. IOW, if he sees “1 ±1”, it’s simply “1-1”. “1–-1” is 1+1. As it was taught to me, you flip up one of the horizontal signs (either minus or negative, whichever works) and make it a positive, since that’s really what it is. A double negative in english might help explain it as well, EG “I didn’t take nothing”, which means “I took something”.

By the by. Have you identified his learning style? If he isn’t suited particularly to visual methods, you might try the two other popular ones. I say this because there’s been a recent (past 5-10 years) surge in trying to identify the various ways people learn best (auditory, visual and haptic/kinetic are the main three, IIRC).

Lastly, I say this because putting it in terms he uses more often might work. Is he a fan of a particular sport (especially football)? If he knows/follows football, talk to him about plays being for a gain or loss. A loss of 5 yards can also be expressed (although poorly, IMNSHO;)) as a gain of negative five yards. Seeing that drawn out (such as a sack or a penalty) might help.