Back To Course6th-8th Grade Math: Practice & Review
55 chapters | 469 lessons
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Jason has taught both College and High School Mathematics and holds a Master's Degree in Math Education.
An integer consists of positive and negative whole numbers, as well as zero. The neat thing about integers is that they are the most common type of number that we see in our everyday lives. Because of this, it is probably important to know some basic rules that dictate how we can use them.
Adding integers is possibly one of the most familiar operations that we know how to do. Most children start their mathematical journey by counting on their fingers. Each finger when counting represents a whole number, which, in turn, represents a small part of our integers!
Sometimes, in life, we are asked to add the same integer a repeated number of times. For example, 2 + 2 + 2 + 2 + 2 = ? We can go ahead and add these up in parts, 2 + 2 = 4 + 2 = 6 + 2 = 8 + 2 = 10, or we can use another method!
Looking at our example of 2 + 2 + 2 + 2 + 2 = ?, we can start to look for a pattern. For example, how many 2s are in this equation? I count 5. So we could say that we are adding 2 five times. In math, we write this as 2 X 5. The 'X' in this equation represents the mathematical operator 'multiply' or 'times'. This means that we are adding 2 to itself 5 times.
2 X 5 = 2 + 2 + 2 + 2 + 2, which equals 10. Therefore, 2 X 5 = 10. Let's try some others!
3 X 2 = 3 + 3, which equals 6
4 X 6 = 4 + 4 + 4 + 4 + 4 + 4, which equals 24
10 X 4 = 10 + 10 + 10 + 10, which equals 40
Of course, we do not want to have to write out the addition counterpart to every multiplication problem, especially if we get something like 10 X 500 or 123 X 632. I don't know about you, but I don't want to have to write out 10 five hundred times! Because of this, we can use the method of multiplication to make our job simpler.
Multiplication is part memorization as much as it is understanding. So yes, we know that 2 X 5 = 2 + 2 + 2 + 2 + 2 = 10, but how can we know that 2 X 5 = 10 without the addition part? What about 3 X 6 = 18 or 7 X 7 = 49? The answer is yes! But we have to work on it. To start, let's look at a multiplication table:
A multiplication table usually has 12 or 13 integers going across the top and 12 or 13 going down the side. Say we wanted to know what 5 X 3 is. We go across the top till we hit the number 5 and then we go down until we hit the number 3. Whatever square we reach is our answer.
Try using the table to find what 5 X 0 is.
Do you get 0 as your answer? What about 6 X 0 or 12 X 0? Notice something strange? Zero is the easiest integer to multiply by because anything times 0 is 0! Likewise, 0 times anything is also 0!
Notice any other strange occurrences? What about when multiplying by 1? 12 X 1 = 12, 11 X 1 = 11, 10 X 1 = 10, 9 X 1 = 9, and so on. When multiplying anything by the integer 1, you get the number that you multiplied by. One is what is known as the multiplicative identity. This means that when multiplying with 1, nothing changes!
As you can see, it is pretty easy to multiply by the first 13 integers (1 - 12 and 0), we can just look it up on our table until we have them memorized. What happens if we reach numbers not on our table? What if we are multiplying two 2-digit numbers together, like 12 X 13? Sure, we could write 12 + 12 + 12 + … + 12 thirteen times, but that could get daunting. Instead, we use a technique called column multiplication. Here's how it works.
First, stack the two numbers on top of each other, making sure each place value lines up.
Second, we are going to start with the ones place of the bottom number (the 3 in this example) and multiply it to each digit in the top number, like so:
Notice how when multiplying along the blue line we get the blue number 6, and when multiplying along the red line we get the red number 3. Once every digit of the top number has been used, we move onto the third step.
Third, we are going multiply by the tens digit of the bottom number; this number is now highlighted in green.
This is where the catch comes in. To multiply by the tens digit, we have to throw in a 0 for a place holder. This works for every digit greater than the ones place. If multiplying by a number in the hundreds digit, we would put two zeros in. This new zero is also highlighted.
Also take notice how our new set of numbers is going to be written directly under the previous set that we just found. Again, follow the red and blue multiplication lines. Start by multiplying the 1 to the ones place in the top number and the tens place.
Once we have multiplied by all place values of the bottom number, we move onto the last step. Step 4 is to add down each column of new numbers, just like doing our normal addition.
So 12 X 13 = 156. Here are a couple more examples:
How about 121 x 137?
Notice in the red example, the extra blue digit that was moved to the top of the 100s place. When you multiply two numbers together in column multiplication, we cannot have anything bigger than 9. If we multiply and get 14, we have to carry the 10s digit, in this case the one, over to the next place value. It's very important to remember that we multiply to the 100s digit like normal. Then we add that carried over. So, 7 times 1 equals 7, then we add the blue one that we carried over to get 8. Once this is done, that blue one is gone. We continue multiplying like normal. Go down a line, add a zero, and start multiplying by 3.
Now we move down another line, add two zeroes, because we are in the hundreds place now, and multiply like before. If we get a number than 9, remember to carry the 10s digit.
Now that all of our place values on the bottom number have been used, we go through an add down again. 121 x 137 equals 847 plus 3,630 plus 12,100, which equals 16,577.
The last hitch with multiplying integers is that they include the negative numbers as well. But this does not have to be a problem! Just follow two simple rules.
When multiplying positive and negative numbers together, add up the number of negative signs you see. If you get an odd number, your answer will be negative and if you get an even number of signs, your answer will be positive.
Example 1: (negative) X (negative) = (positive)
Example 2: (negative) X (positive) = (negative)
Example 3: (negative) X (negative) X (negative) X (positive) has an odd number of negatives (3 negatives), so the answer will be negative.
Multiplying the actual numbers involved works the same way as all of the above examples. The only thing that changes is the sign.
Example 4: -3 X 2 = -6
Example 5: -3 X -10 = 30
Example 7: -4 X 5 X 2 X -3 = 120
To do this last example, multiply -4 X 5 to get -20. Then, multiply -20 X 2 to get -40. Lastly, multiply -40 X -3 to get 120.
Multiplying integers is the same thing as adding a number a certain number of times.
5 X 3 = 5 + 5 + 5 = 15
Once numbers get large, we can use multiplication tables and column multiplication to help us find our answers. When multiplying more than one number together, just work two at a time until every number is used. Lastly, if we have negative numbers in our problems, we can simply count the number of negatives that are there. If we have an odd number then our answer is negative, and if we have an even number of negative signs, then our answer is positive!
When you are finished, you should be able to multiply positive and negative integers together to find a product.
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Back To Course6th-8th Grade Math: Practice & Review
55 chapters | 469 lessons