Susan Johnsey - online math courses and math tutoring

Friday, October 29, 2010

Adding and subtracting integers (or any real number) can be a very daunting task for those beginning algebra or prealgebra. Most students will understand the rules for adding signed numbers, but once introduced to subtraction that are quite confused. The multiplication and division rules for signed numbers are usually quite easy for the students to master. The addition and subtraction rules are quite confusing for many and thus can prevent a student from be successfull in their first semester of Algebra.

So what do we do??? I was teaching students with learning disabilities and decided that I had to create a way to present addition and subtraction in a simpler manner and definitely not with the normal "math" rules presented in textbooks. Our budget was also low so I could not use manuipulatives; I also did not want to have to keep up with them.

Unteaching a topic can be tough because there are usually rules that the student has misunderstood or only "half-way" remember. Thus I do have to ask students that already have been taught the school rules, but are not succesful at using them to please try to forget those and just think about what I am presenting. After we complete a few problems they will see how it all works together.

Here is my way of teaching addition and subtraction of signed numbers. Please note that I will teach these all at once and will not really focus on whether we are subtracting or adding!! And yes, teachers the kids get it.

All numbers have a sign.
IT is always on the left side of the number.

-2 + 4 There is a negative 2 and a positive 4.
3 - 5 There is a positive 3 and a negative 5.
- 5 - 8 There is a negative 5 and a negative 8.

Do you see the sign is on the left always.
If no sign is written then it is positive. That is why the 3 is positive.

2. What if there are 2 signs on the left?

+-7 or +(-7) or -(+7) are all = -7.
We can ignore "extra" positives sometimes, but never negatives!!

- (-6) or - - 6 are the same number. They both mean Positive 6.

“Double negatives”, that is a negative sign followed immediately
by another negative sign yields one positive.

[Note: I did not say one negative number followed by another negative number,
such as -7-8. That is not what I am explaining here. The 7 is negative and the 8 is negative.]

7- (-6) becomes 7 +6 The “double negatives” disappear and become one positive. Why is this true? A little logic will explain that. Positive and negatives are opposites of each other. Right? yes. 3 and -3 are opposites. The opposite of a -6 is a +6. Another way to think of negatives is to think the word opposite. -2 can be called negative 2, minus 2, or opposite of 2.

Thus the OPPOSITE of -6 is a positive 6 can be written as a math expression:
-( -6) = +6.

So we know -5 - (-3) becomes -5 +3 .
1. Every number has a sign and it is always on its left side.
2. We know by logic that "double negatives" yield a positive.

How do we add or subtract signed numbers?
Create or draw 2 boxes. One is for the negatives and one for positives.


Now that we know about the signs, let’s use the positive and negative boxes to explain adding and subtracting of integers. Most books teach this in 2 lessons, first adding and then subtracting. I do not. When we are “adding integers” there are times when we actually subtract the numbers!! And when “subtracting integers” there are times when we add the numbers.!! Confused? Yes ? no? Please read on. First, lose your “mind set” about this topic from any prior knowledge you may have. This is simple.
Warning: DO NOT write the signs inside the box.

-5 + 3 or 3 - 5 both mean

a 5 is placed in the negative box and 3 is placed in the positive box.

Remember, the sign of the number is on its left. If a number has no sign on its left then it is positive.

In both expressions above the 5 is negative and the 3 is positive. Now we ask these 2 questions.


1. which box has more in it? DO NOT write the signs inside the box.
2. how much more does it have?

For -5 +3 or 3 - 5 the negative box has more and it has 2 more than the positive box.

Our answer is thus negative 2. Thus -5 +3 = -2.
We have more negatives. The negative box has 2 more than the positive box.
-5 +3 or 3 - 5 are equal to -2.

-2 - 4 means you have 2 in the negative box and then 4 more are placed into the negative box.

Which box has more?
The negative box.
How much more? Since the positive box has none and the negative box has 6 then obviously the negative box has 6 more. The answer is -6. that is, -2 - 4 = - 6

- 4 +4 means 4 is placed in the negative box and 4 is placed in the positive box.
If the amounts in the 2 boxes are equal then your answer is 0, ZERO
Also -3+3 =0 can be written +3 - 3=0 or 3 - 3 =0.
Remember the sign of a number is on its left.

-12 + 4 means you have 12 in the negative box and 4 in the positive box .

Which box has more? The negative box has 8 more. The answer is - 8. That’s -12+ 4 = - 8.

This is the same problem as +4 - 12. Be sure you see that. Just look at the signs.

What about when there are double signs?

-2 - +3 or
-2 - (+3) or
-2 + (- 3)
Recall I suggest ignoring the extra positives. I said extra positives. We can never ignore negatives. All three of these become - 2 - 3. Cover the extra positive with your pencil.
The above problems are the same as -2 - 3. We have 2 in the negative box and another 3 is placed into the negative box . The answer is - 5.

What about double negative signs?
- -4 is an example . Think of this as the opposite of negative 4. The opposite of negative is, of course, positive. - - 4 is simply +4. or -(-4) = +4 or 4

- -4 +6 becomes +4+6. This is also printed as - (-4) +6 and it becomes 4+6.
Both are in the positive box so our answer is +10 or just 10.

-7- (-8) becomes -7 +8 . LOOK at that. The double negative was on the 8. The sign of a number is always on its left. -7 +8 = +1. There is one more in the positive box than in the negative.

WARNING: DO NOT USE the boxes until you have changed your "double signs".

-7- (-8) = -7 +8 = 1

I know these are not the usual math rules. But if you are struggling then forget what you know and try these. They work. Using the other stuff you sort of know can really get in the way. Now go back and do this, please:

Write all of the above examples with their boxes on a card for reference.

-8+8 = 0
The boxes have the same amount so neither one has more. Answer is 0.

There are rules also for multiplying and dividing, but they are easy. A positive number times a negative number is always a negative number. A negative number times another negative number always equals a positive number. These 2 rules apply to division also, BUT not adding and subtracting!!! ( Use your box for adding and subtracting.)

Compute and email you answers. It is a free service or check below.

1. -6+9
2. -6 -9
3. -6 -(-6)
4. 6- 9
5. 6 -(-9)

6. 8 - (-5)
7. 8 - (5)
8. -8 +8
9. -5 -8
10. -5 -(-8)

Susan Johnsey

11. -7 -9
12. -8 +10
13. 14-(-8)
14. 8+(-9)
15. 8-(-9)

16. -9 +7
17. -8 -10
18. 8 - 10
19. -8 +(-9)
20. 9-(-6)

21. 7 - 9
22. -10 +8
23. -10 -8
24. -9-8
25. -6+9