Multiplication and Division


We'll begin this lesson by talking about the basics of multiplication. Hold up 3 fingers on one hand. Now hold up 3 fingers on the other hand. You should have 2 hands with 3 fingers being held up on each. What does this tell us? It tells us that 2x3=6. 2 hands x 3 fingers held up on each hand = a total of 6 fingers being held up. Now let's try something else. Take a piece of paper and draw 4 lines. Put a circle around these 4 lines. Now do the same thing 2 more times. You should have 3 circles with 4 lines in each of them. We could say that 4+4+4=12, but there's another way to do it. We know that we have 3 circles. Each circle has 4 lines inside of it. It's easier to say that 3x4=12. 3 groups x 4 lines in each group = a total of 12 lines altogether. See how easy it is to multiply? Take a look at the picture below. It shows some other examples of multiplication.


Now let's talk about division. Division is the opposite of multiplication. Normally with multiplication you are taking a small group of something and multiplying it into a larger group made up of those small groups. Most often with division you are taking a larger group of something and dividing it into smaller groups to find how many of those small groups are contained within the larger group. Of course with math we are normally working with numbers instead of groups, but it often helps to think of it this way.

Stick those same 6 fingers up again, 3 fingers on one hand and 3 fingers on the other. Hold them close together. You have a group of 6 fingers. Now hold them as far apart as you can. You just divided the group of 6 fingers into smaller groups of 3 and found that there were 2 groups of 3 fingers contained within the larger group of 6 fingers. In other words, 6/3=2.

Let's try something else. Take a piece of paper and draw 12 lines on it. Now put a circle around each group of 4 lines. When you're finished, you should have 3 circles with 4 lines inside of each of them. What we just did was divide the group of 12 lines into smaller groups of 4 lines. We found that there were 3 groups of 4 lines contained within the larger group of 12 lines. In other words, 12/4=3. Please take a look at the picture below for some other examples of division.


To check your answer when you multiply, to see if it is right, you have to reverse what you did and divide. Take the answer you got, divide by the middle number, and the result should be what you started with. To check your answer when you divide, to see if it is right, you also have to reverse what you did, but this time you multiply. Take the answer you got, multiply by the middle number, and the result should be what you started with. Below is a picture that gives an example of what I'm talking about.


Problem: 5x2=10
To check it, reverse things and divide.
Check: 10/2=5
5 is what we started with so this must be right!

Problem: 6/3=2
To check it, reverse things and multiply.
Check: 2x3=6
6 is what we started with so this must be right!

Multiplication gets a little trickier with larger numbers. Having your multiplication tables memorized helps a lot. With addition and subtraction we can cheat sometimes and use our fingers and toes to count with. It's kind of hard to cheat when we have a problem like 9x9 though! I have a picture below that shows the steps in a multiplication problem with larger numbers. Look the picture over, and then I'll explain each step. Notice how the problem is set up. We scooted the numbers over to the right. The problem is 45x27. I don't know about other people, but the multiplication tables I learned only went up to 12! For big numbers like this we need either a calculator, or the knowledge to work these problems without one. Since we won't always have a calculator handy, we need to know how to work these large problems on our own. Let's look at the problem.


In step 1 above we have to begin working the multiplication problem with the furthest number to the right on the bottom row. We have to multiply it by the number above it. In step 2 I multiplied 7x5 and found that it equaled 35. Since we can only write one number below the 7, I had to "carry" the 3 and put it over the column of numbers to the left. I wrote the 5 directly below the 7. In step 3 I haven't finished with the 7 yet. Now I have to multiply it by the other number on top, which is the 4. 7x4=28. Since I had to carry the 3 from step 2, I'll add it to my answer. 28+3=31. Since there are no other numbers to the left of the 4, I can simply write the 31 down directly below the column where the 4 is. If there had been another number, I would have had to carry the 3 again, and repeat the same process over for that number.

In step 4 we're now done with the 7 so we can forget about it. We move over one position to the left on the bottom row and find a 2. We have to multiply this 2 by everything on the top, just like we did with the 7. We start with the number furthest to the right on top again, which is 5. 2x5=10. Again, we can only write one number below the 2. Since our answer was 10, we write the 0 below the 2 on a new row below the one we used for the 7 and carry the 1, putting it over the next column of numbers to the left.


In step 5 we move over to the next number to the left on top. This number is 4. 2x4=8. Since we had to carry the 1, we add it to our answer. 8+1=9. We write the 9 to the left of the 0 we just wrote down in step 4. Now we're done with the 2. There are no more numbers left to multiply so all we have to do now is to add everything up that we have below the line and see what our answer is. In step 6 we add 315+900. Wait a minute you say! The picture says 315+90. Let me explain. When you wrote the numbers down under the line, they had to be in certain positions for the answer to be right. Take the number 315 for instance. 5 is in the "ones" position. 1 is in the "tens" position. 3 is in the "hundreds" position. Since the 9 is right below the 3, it is also in the hundreds position. If it helps, you could write in an "imaginary" 0 to the right of the 90. There's nothing wrong with that, and to be honest, it's actually there, we just don't "see" it. With that said, 315+900=1215. Our final answer is 1215!

Division is also a little trickier with larger numbers. We set up the problem up as shown in the picture below. The problem we will work is 1024/11. You think it's too hard? No way! I'll show you just how easy division is with large numbers! The main thing to remember is keeping everything lined up right. Notice where we put the 11 and where we put the 1024. This is the way to set up a long division problem. Take a look at the picture, then I'll explain how we work the problem step by step.


In step 1 above we check to see if we want to divide the first two numbers by 11. 10/11 doesn't work, so we'll move over another number and see if we want to divide 102 by 11. This time it does work, because 102/11 doesn't give us any problems! In step 2 I divided 102 by 11 and got 9 with a remainder of 3 for my answer. I can't do anything with the remainder right now so I just forgot about it. I wrote the 9 directly over the 2, since that was the last digit of the number I chose to start working with. Next I multiplied 9x11 and got 99. I wrote the 99 directly below the 102, lining up things from right to left. In step 3 I subtracted 99 from 102 and got 3. (3 was the remainder of 102/11 if you remember.) Since 3/11 doesn't work, I brought down the 4 from 1024. 34/11 does work!


In step 4 I divided 34 by 11 and got 3 with a remainder of 1. I still can't do anything with the remainder so I just forgot about it again. I wrote the 3 to the right of the 9 on top. Next I multiplied 3x11 and got 33. I wrote the 33 directly below the 34, lining up things from right to left. In step 5 I ran out of numbers to divide by 11. Now the only thing that was left to do was to subtract 33 from 34. 34-33=1. (If you remember, 1 was the remainder of 34/11.) Since we don't have anything left to divide by, our final answer is 1024/11=93 with a remainder of 1! You can also write the remainder as a fraction like this 1/11, but for now, we usually just write it as "r1" ("r" stands for "remainder").

I'd like to point out an interesting rule for multiplication. With multiplication, it doesn't matter what order the numbers are in; the answer will be the same. This is NOT true for division though! When dividing, the numbers must always remain in the exact order that they are in or your answer will be wrong! Below are some examples of what I am talking about.

*** Multiplication ***

(is the same as)

(is the same as)

*** Division ***

(is NOT the same as)

(is NOT the same as)

You made it! We're through with the lesson! That wasn't so bad was it? Now's a good time to scroll back up the page and make sure you understand everything. If you feel weak in a certain area, try doing a few practice problems on your own until you're stronger. You'll find that this stuff is pretty easy after you work a few problems. Below is a quiz to see how much you've learned. You don't have to take it of course, but why not? No one's going to see what you make but you. Besides, it'll help you figure out where you might need to practice more. Good luck and thanks for reading through the lesson!


Question 1:

1 x 2 = _____

A:  1
B:  2
C:  3
D:  4

Question 2:

2 x 4 = _____

A:  2
B:  4
C:  6
D:  8

Question 3:

3 x 5 = _____

A:  8
B:  10
C:  12
D:  15

Question 4:

2 x 2 x 3 = _____

A:  4
B:  6
C:  7
D:  12

Question 5:

15 x 10 = _____

A:  15
B:  100
C:  150
D:  1510

Question 6:

2 / 1 = _____

A:  0
B:  1
C:  2
D:  3

Question 7:

6 / 2 = _____

A:  2
B:  3
C:  4
D:  6

Question 8:

8 / 4 = _____

A:  2
B:  4
C:  8
D:  12

Question 9:

10 / 2 = _____

A:  1
B:  5
C:  8
D:  10

Question 10:

80 / 4 = _____

A:  20
B:  40
C:  60
D:  80

~ Other Links ~

Multiplication Mystery

Captain Nick Knack

Whole Numbers and Their Basic Properties

Let's Do Math!

Coolmath4kids - Multiplication and Division

Coolmath4kids - Long Division

Harcourt E-Lab

Harcourt Math Glossary

Ask Dr. Math Multiplication

Ask Dr. Math Division

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