COMPUTING WITH INTEGERS


Computing with integers is an important skill for higher level algebra work. Here are the ways we have learned to compute with integers in class...

Adding on a Number Line
Adding with Integer Chips
Adding by Thinking about Integer Chips
The Rule for Adding Integers

Subtracting with Integer Chips
Subtracting on a Number Line
The Rule for Changing Subtraction Problems

The Rule for Multiplying Integers

The Rule for Dividing Integers





Adding on a Number Line
Working on a number line makes adding integers visual! Suppose we want to add 5 + –9.

Step 1: Start at zero
Always start at zero when working on a number line.

Step 2: Draw an arrow to represent the first number
Since the first number is a positive 5, we start at zero and move 5 steps in the positive direction. Draw the arrow from zero to the ending point.

Step 3: From there, draw a second arrow to represent the second number
The second number is -9, so we now move 9 steps in the negative direction starting from where we left off. Draw the arrow to the new ending point.

Step 4: Look at the number line for the answer
The place on the number line where you end up is the answer. Since we ended up at –4, we know that 5 + –9 = –4.






Adding with Integer Chips
Working with integer chips makes adding integers visual, as well as hands-on! Suppose we want to add –6 + 8.
Each red chip represents a negative 1. Each black chip represents a positive 1.

Step 1: Take out enough chips to represent the first number.
In our case, the first number is –6. We need six red chips.

Step 2: Take out enough chips to represent the second number, then combine them.
In our case, the second number is positive 8. We need 8 black chips.

Step 3: Make zero-pairs
We know that one positive and one negative combine to make zero, so we call this a zero-pair (a pair of chips that combine to make zero).
Make as many zero-pairs as you can with your chips. You know you are done when you can't make any more zero pairs - you will either run out of positive chips or negative chips.

Step 4: The chips left over make your answer
Since all of the zero-pairs combine to make zero, count up the chips that are not part of a zero-pair. If the left over chips are negative chips, your answer is negative. If the left over chips are positive, your answer is positive. This is your answer!
In our case, the answer is 2 because there are only 2 positive chips left.





Adding by Thinking About Integer Chips

Now that you are an expert with adding integer chips, you can add with integer chips mentally!
Suppose we want to add –650 + 800.

Step 1: Will the answer be positive or negative?
Do you have more positive chips or negative chips? Whichever you have more of, your answer will have that sign.
In our example, we have 800 positive chips and 650 negative chips. Since we have more positive chips, the answer will be positive.

Step 2: How many zero-pairs will there be?
You know that you are going to make zero-pairs until you run out of either positive chips or negative chips, so which one will you run out of first?
In our case, since we have less negative chips, we know we will run out of negative chips first. Therefore we will be able to make 650 zero-pairs.

Step 3: What chips will be left after making the zero-pairs?
After all of the zero-pairs are made, what is left?
In our case, after we've made 650 zero-pairs, we will only have 150 positive chips left.

Step 4: What is the answer?
Whatever is left is the answer!
In our case, we have 150 positive chips left so our answer is 150!





The Rule for Adding Integers

If you are the type of person who needs a rule to follow, here it is!

If the signs are the same (both positive or both negative)...
Ignore the signs and add the two numbers together. Then put the original sign back on your answer.

For example, –6 + –8. We first ignore the signs and add, so 6 + 8 = 14. Since the original signs were negative, I take that sign. So my answer is –14.


If the signs are different (one is positive and one is negative)...
Ignore the signs and subtract the numbers. Look to see if you have more positive chips or negative chips, and take that sign and put it on your answer.

For example, –9 + 11. We first ignore the signs and subtract, 11 – 9 = 2. Since we have more positive chips than negative (we started with 11 positive and 9 negative), take that sign. So my answer is +2.





Subtracting with Integer Chips

Working with integer chips makes adding integers visual, as well as hands-on! Suppose we want to subtract 4 – –3.
Each red chip represents a negative 1. Each black chip represents a positive 1.

Step 1: Take out enough chips to represent the first number
In our case, the first number is positive 4. We need 4 black chips.

Step 2: Add in zero-pairs to the original number
Do we have enough chips to take away the second number? If not, you are going to have to put zero-pairs into the original number so that there are enough chips to take away.
In our case, since we are going to have to eventually take away 3 negative chips, we need 3 negative chips to take away! Add 3 zero-pairs.
Notice how adding zero-pairs doesn't change the value of the original number, it just puts more chips on the board.

Step 3: Take away the second number
Subtraction means to take away, so take away the second number from the chip board.
In our case, we are taking away 3 negative chips.

Step 4: The chips left over make your answer
Since we have 7 positive chips left, our answer is positive 7.



More Examples to Understand
2 – –5 = 7


–4 – 2 = –6


–3 – –7 = 4







Subtracting on a Number Line

Working on a number line makes subtracting integers visual! Suppose we want to subtract –3 – –6.

Step 1: Start at zero
Always start at zero when working on a number line.

Step 2: Draw an arrow to represent the first number
Since the first number is a negative 3, we start at zero and move 3 steps in the negative direction. Draw the arrow from zero to the ending point.

Step 3: Draw a second arrow to represent the second number but in the OPPOSITE DIRECTION
The second number is –6, so we are going to move 6 steps. But because we are subtracting, we are going to go in the opposite direction that it tells us. Since we have a negative 6, we are going to move 6 steps in the positive directions! Draw the arrow to the new ending point.

Step 4: Look at the number line for the answer
The place on the number line where you end up is the answer. Since we ended up at 3, we know that –3 – –6 = 3.



More Examples to Understand
2 – –5 = 7


–4 – 2 = –6


–3 – –7 = 4





The Rule for Changing Subtraction Problems

You can change any subtraction problem into an addition problem! Since we know how to solve addition problems easily, we can then solve the problem!

Step 1: Keep the First Number the Same
Always keep the first number in a subtraction problem exactly the same.

Step 2: Add the Opposite of the Second Number
Just like with the number lines, we take the opposite of the second number when we subtract. This makes an addition problem that we know how to solve!

Examples:
–4 – 5 = –4 + –5
–8 – –3 = –8 + 3
10 – –12 = 10 + 12





The Rule for Multiplying Integers

To multiply two integers, follow this rule...

If the signs are the same (both positive or both negative)...
The answer will be positive. Ignore the signs and multiply the numbers together. The answer will be the positive answer. For example, –6 x –8. Since they are both negative, the answer will be positive. Ignore the signs and multiply, 6 x 8 = 48. The answer will be positive 48.


If the signs are different (one is positive and one is negative)...
The answer will be negative. Ignore the signs and multiply the numbers together. The answer will be the negative answer. For example, –9 x 7. Since the signs are different, the answer will be negative. Ignore the signs and multiply, 9 x 7 = 63. The answer will be –63.





The Rule for Dividing Integers

To divide two integers, follow this rule...

If the signs are the same (both positive or both negative)...
The answer will be positive. Ignore the signs and divide the numbers. The answer will be the positive answer. For example, –54 ÷ –6. Since they are both negative, the answer will be positive. Ignore the signs and divide, 54 ÷ 6 = 9. The answer will be positive 9.


If the signs are different (one is positive and one is negative)...
The answer will be negative. Ignore the signs and divide. The answer will be the negative answer. For example, –18 ÷ 2. Since the signs are different, the answer will be negative. Ignore the signs and divide, 18 ÷ 2 = 9. The answer will be –9.



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Copyright © 2004 Ricky Ohashi