The Math Files
Unit 1: Rational Numbers
Article: Adding and Subtracting Integers
Learning Targets:
I can…
o Add, subtract, multiply and divide integers
o I can also model these operations
o Use the Order of Operations to complete problems
Math is a complicated field of study. Each year, the concepts broaden and become more complicated, but the essence of
math is in the relationships between numbers. These relationships are objective, meaning they are always true and do not
change. Our first unit this year in Math 7 is meant to help us describe these complicated relationships.
We begin the year with integers. Integers include all whole numbers and their opposites. More simply, they are
positive and negative numbers.
We see negative numbers all around us in real life. We see it in temperatures, like when the weatherman describes
yesterday’s temperature as -5. We see it with money or finance, like if I overpay the credit card bill, and our credit card
balance is $-50. We even see it in height, like when the submarine was 100 feet BELOW sea level, or when the parking
garage is below the lobby and the elevator has numbers below the L. I would be willing to bet that you could come up with
even more examples if you thought about it!
Negative numbers represent numbers that are less than zero. They are below 0, and the greater the absolute value of
a negative number is, the smaller that number is. What I mean, more simply, is – the larger the negative, the smaller
the number. We know, for example, that -50 degrees is colder than -25 degrees. We also know that 100 feet below the
surface is deeper than 40 feet below the surface. Negative numbers help us see these relationships.
There are a few operations with negatives that we need to talk about. The first operation is addition. Adding a negative
number will always make the sum smaller than the original number. For instance if I add 6 + (-5). It is like I start 6 feet in
the air, and fall back five feet toward the ground. I have added a negative to a positive, and now I am only 1 foot above
the ground. Therefore 6 + (-5) = 1. When I add a negative number, the sum will always decrease. If I start as a negative
and add another negative, the sum will be lower. In (-15) + (-10), it is like I am 15 feet below the surface of the water, and
I swim down another ten feet to see the coral. I am now 25 feet below the surface. I am now at -25. I have descended
deeper below the ocean.
In order to simplify the process in addition, I have created two, three word phrases to remember.
SAME
OPPOSITE
ADD
SUBTRACT
KEEP
BIG
1. Same – In the example above, both integers have the
same sign. They are both negative
2. Add – I ignore the negative sign momentarily, and add
the two numbers together. 5 + 12 is 17.
3. Keep – I write down my answer as 17, and notice that
both signs were negative in the original problem. So, I
keep the negative sign at the end, and the sum of -5
and -12 become NEGATIVE 17.
1. Opposite – In the example above, the two integers
have opposite signs (Negative seventeen plus
POSITIVE thirteen.) One is positive and one is
negative.
2. Subtract – I ignore the negative signs momentarily,
and subtract the bigger number (the number with a
greater absolute value) minus the smaller one.
3. Big – Once I find my difference (in this case 17 – 13 is
4), I determine which number has a bigger absolute
value. My answer will have the sign of that number.
17 is greater than 13, and 17 is negative. Therefore my
answer is NEGATIVE 4.
Adding integers is a simple process. The mistake most students make is forgetting to put a negative sign back on their
answer. It brings us to an important phrase which will frequently be repeated this year: “Don’t lose the negative!”
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The Math Files
Subtraction can be more difficult. Subtracting a positive number will always make the difference smaller than the original
number. However, subtracting a negative number will make the difference larger than the original number. It is like if you
owe $500, and I take away (a layman’s term for subtraction) your debt. I have subtracted the negative amount of money,
and now you owe $0.
This can be complicated, but in order to simplify the process, we have concocted a second three-word phrase. This
phrase will help you turn all of your subtraction into addition. The phrase is Keep Change Change. Once you turn the
subtraction into addition, you must follow the adding rules above.
KEEP
CHANGE
CHANGE
1.
2.
3.
4.
5.
6.
Keep – Keep the first number exactly the same. -15 stays as negative 15.
Change – Change the sign to addition. Minus becomes plus.
Change – Change the sign of the last number. 25 becomes -25.
Follow your addition rules.
These have the same sign, so we follow SAME ADD KEEP.
Our answer is -40.
Integer operations are fundamental to understanding all of the math that follows. This is a foundational skill! We build the
whole year on this understanding. These three word phrases are merely tricks that I have discovered to make the process
easier to learn. There are other ways to learn this material, but this is designed to help!
Questions:
1. What are integers?
2. Where are some places that you have seen integers in your everyday life?
3. What three word phrase would we use with the problem -7 + -4? How do you know?
4. What is -7 + -4?
5. What three word phrase will turn all subtraction problems into addition?