0-3 Subtract Integers

Learning Goal: I can subtract integers.

Examples:

There are six different scenarios that might occur as follows—

Larger positive integer minus smaller positive integer

+ - + = +

5 – 3 = 2

or +5 – (+3) = +2

Smaller positive integer minus larger positive integer

+ - + = -

4 – 9 = -5

or +4 – (+9) = -5

Positive integer minus negative integer

+ - - = +

7 - -1 = 8

or 7 – (-1) = +8

Negative integer minus positive integer

- - + = -

-2 – 9 = -11

or -2 – (+9) = -11

Negative integer further from zero (highest absolute value) minus negative integer closer to zero (lowest absolute value)

- - - = -

-10 – -3 = -7

or -10 – (-3) = -7

Negative integer closer to zero (lowest absolute value) minus negative integer further from zero (highest absolute value)

- - - = +

-2 - -6 = +4

or -2 – (-6) = +4

Critical Importance Level: 4 stars

There is nothing more important to learn in math than this. Without mastery of this skill, you can never again succeed in math.

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Vocabulary

What is the additive inverse?

The additive inverse is the opposite number. The opposite number is the number that you add to a number so the sum is zero.

An example of opposite numbers: 6 is positive 6. The opposite of positive 6 is negative 6.+6 + (-6 ) = 0So -6 is the opposite or additive inverse of +6.

-52 is negative 52. Add positive 52 to negative 52. The answer is 0.-52 + (+52) = 0So +52 is the opposite or additive inverse of -52.

Zero Pairs

Some people call a number plus its opposite number zero pairs. You will hear that term on a lot of the videos. Zero pairs are two numbers that are opposites of each other, and that equal zero when added together.

Target skills:

Subtract IntegersEvaluate Algebraic Expressions

There are several methods for subtracting integers.

Adding the additive inverse (textbook way)Yellow and red counters (add the additive inverse)Number line Boxing the integers and summing them

Adding the additive inverse rules (abstract)

To subtract integers, follow this rule:

1. Instead of subtracting, add the additive inverse.

To do this, follow these steps:A. Change the minus sign to a plus sign.B. Change the second number to its opposite number.If it was +3, make it - 3.If it was - 62, make it + 62.C. Then, rewrite the problem.D. Now, follow the rules for adding integers.

Some subtracting integer examples:

A positive number minus a positive number.

+ - +

5 - 3

This is the same as 5 - (+3)

Add the additive inverse. 1. The additive inverse of +3 = -3 How? A. Change the minus sign to a plus sign 5 + (+3) B. Now change (+3) to its additive inverse or its opposite number.5 + (-3).

C. Now you have made what was a subtraction problem into an addition problem.D. Follow the rules for adding integers from this point on.

5 + (-3) When adding two numbers with different signs. 1. Subtract the numbers: 5 - 3 = 2 2. Keep the sign of the largest number. 5 is greater than 3. 5 is positive. Final answer = +2 Or you could do that problem the same way you have all along:

5 - 3 = 2

Another subtracting example:

A negative number minus a negative number.

-7 - (-4)

1. Add the additive inverse. The additive inverse of -4 = +4 How? A. Change the minus sign to a plus sign -7 + (-4)

B. Now change (-4) to its additive inverse or its opposite number.-7 + (+4).C. Now you have made what was a subtraction problem into an addition problem.D. Follow the rules for adding integers from this point on.

-7 + (+4) When adding two numbers with different signs. 1. Subtract the numbers:7 - 4 = 3 2. Keep the sign of the largest number. 7 is greater than 4. 7 is negative. Final answer = -3

Another subtracting example.

A positive number minus a negative number.

+ - -

13 - - 9

Usually written like this: 13 - (-9)

13 - (-9)

1. Add the additive inverse. 2. The additive inverse of -9 = +9 How? A. Change the minus sign to a plus sign 13 + (-9) B. Now change (-9) to its additive inverse or its opposite number.13 + (+9).C. Now you have made what was a subtraction problem into an addition problem.D. Follow the rules for adding integers from this point on.

13 + (+9) When adding two numbers with the same signs. 1. Add the numbers: 13 + 9 = 22 2. Keep the sign. Both numbers, 13 and 9, are positive. Final answer = + 22

On problems like this where there is this situation A number minus a negative number - (- ) I actually think of it like this: a negative times a negative equals a positive.

Another subtraction example.

A negative number minus a positive number. - - + -7 - (+45)-7 - (+45) 1. Add the additive inverse. The additive inverse of +45 = -45 How? A. Change the minus sign to a plus sign -7 + (+45) B. Now change (+45) to its additive inverse or its opposite number. -7 + (-45).C. Now you have made what was a subtraction problem into an addition problem.D. Follow the rules for adding integers from this point on.

-7 + (-45)

When adding two numbers with the same signs. 1. Add the numbers: 7 + 45 = 52 2. Keep the sign. Both numbers, 7 and 45, are negative. Final answer = -52

Another way: Yellow and Red Counters (concrete)

1. Create a pile for each integer in the problem.Note: The piles must be arranged in the same order they appear in the problem.-6 - +2 =oooooo – oo =2. Change the color of the second pile of counters to the

opposite color.

oooooo – oo =

3. Join the two piles together. Yes, you are adding, even though this is subtracting. Crazy, I know.

oooooooo

4. If necessary, remove zero pairs. This is necessary if the joined pile has both red and yellow counters. Remove one red and one yellow counter at the same time, until only one color of counter remains in the pile.

5. The final answer equals the number of counters left in the pile. The sign is determined by the color of the counters in the final answer pile.

Remember: a yellow counter = +1 and a red counter = -1

The final answer to -6 - +2 = -8

The number line solution (representational)

1. Start at 0 on the number line.2. If the first integer is positive, move to the right on

the number line the number of units indicated by the digit.If the first digit is negative, move to the left on the number line the number of units indicated by the digit.

3. From this landing spot, go the opposite way that the next integer indicates.If the integer is positive-move to the left instead of the right.If the integer is negative-move to the right instead of the left.

Note: If you are solving a problem with more than one minus operation, start at the beginning, and move on the number line taking into consideration one integer at a time. After the first integer in the problem, always move

the opposite direction on the number line than is indicated by the integer’s positive or negative sign.

The “box” way.

This is the way I personally add and subtract integers.

It is very similar to the way described here:http://www.mathinabox.com/AddSubIntegers/Adding_and_subtracting_integers.htm

However, I follow these steps in my head.

1. If there are two signs stuck together, multiply the signs following the multiplying integers rules.Note: I am not multiplying. I am subtracting. But in order to more easily subtract, I will multiply ONLY the signs stuck together. Always do this step first, if necessary. It is necessary if two signs are stuck together, or if they two signs are separated by a parenthesis symbol between them.

Multiplying Integers Rules

+ times + = +Positive times a positive equals a positive.

- times - = +Negative times a negative equals a positive.

+ times – OR –times + = -Positive times a negative or a negative times a positive equals a negative.

2. Now each integer will have only ONE sign in front of it.3. The sign in front of an integer is the sign that belongs

to the integer. If there is no sign in front of an integer, it is an invisible positive sign. Negative signs are never allowed to be invisible.

4. Try to see in your mind each integer with its sign in a box.

5. Join (add) the amounts together following the rules to add NOT subtract integers.

Example:-7 – (-2) =1. Multiply the two signs stuck together.

-(- = negative times a negative equals a positive.2. Now I have a new problem.-7 + 2My two integers are:-7+2Follow the rules to add integers.1. The signs are different, so I will subtract the numbers

(even though I am now ADDING INTEGERS, even though this was originally a subtraction problem.) Crazy, I know.7 minus 2 = 5 5 is part of the answer.

2. -7 is farthest from zero, so my answer will be negative, because the sign for 7 is negative.

So my final answer is -5.-7- (-2) = -5

Try another problem:

6 – (+1) =

1. Multiply the two signs together.-(+ = a negative times a positive equals a negative.

Change the problem to this:6 – 1 = Note: I can recognize this problem from 1st grade!

6 – 1 = 5.

But following the steps.Now look at the problem this way.+ 6 – 1 =Follow the rules to add integers from here on out.1. The integers have different signs, so subtract the two digits. 6-1 = 5.

5 is part of the answer.2. The 6 is farthest from zero (has the greatest absolute

value), so the sign belonging to the 6 will also belong in the answer. 6 is positive. So the final answer is positive.

+6 – 1 = +5.The final answer is +5