Addition and Subtraction


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It may help to have one of your parents read through this lesson with you. If you don't understand something, they can probably help.

Let's begin by talking about addition. Hold up three fingers on one hand. Now hold up three fingers on the other hand. Count all of the fingers on both hands that you're holding up. If you got 6, you're exactly right! If you didn't, maybe one finger got tired or something! For those of you who got 6, you just added 3+3. You found that 3+3=6. Addition is easy to learn if you just practice a little. If you have 4 toothbrushes and add 4 more, how many toothbrushes do you have? Let's count them and see. On a piece of paper draw 4 lines to represent 4 of the toothbrushes. Draw 4 more lines to represent the other 4 toothbrushes. Now count all of the lines you have drawn. Hopefully you got 8! This tells us that 4+4=8. Take a close look at the picture below and notice how easy it is to add.


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Now let's talk about subtraction. Subtraction is the opposite of addition. Instead of adding things, we're going to take them away. Hold up 3 fingers on one hand. Now hold up 3 fingers on your other hand just like before. You should be holding up a total of 6 fingers. Now close three fingers on one hand. How many fingers are you holding up now? You should only be holding up 3 fingers. This tells us that if we have 6 fingers and take away 3 of them, only 3 are left. In other words, 6-3=3. Take the piece of paper with the 8 lines that represent the toothbrushes. On that paper cross out 4 of those lines. How many lines are left that are not crossed out? The answer should be 4. If you have 8 toothbrushes and you take 4 of them away, you are only going to have 4 toothbrushes left. In other words, 8-4=4. Take a close look at the picture below. Notice how easy it is to subtract.


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To see if your answer is correct when you're adding, you need to turn everything around and subtract. Take your answer and subtract the middle number. The answer you get should be the first number.

To see if your answer is correct when you're subtracting, you need to turn everything around again, but this time you need to add. Take your answer and add the middle number. The answer you get should be the first number again.

Look very closely at the picture below. It will help explain how this works. It sounds a bit confusing at first, but if you try a few problems on your own, you will see that it is really quite simple.


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Problem: 1+2=3
To check it, turn it around and subtract.
Check: 3-2=1
1 is what we started with so this must be right!

Problem: 4-3=1
To check it, turn it around and add.
Check: 1+3=4
4 is what we started with so this must be right!


Addition gets a little trickier with larger numbers, but it's nothing we can't handle. First let's talk about how we want to set up our numbers. For small numbers, we don't have to worry about setting them up. We can just write them straight across like this, 2+9=11. If you have a number larger than 9, then you need to write the numbers on top of each other. Take each number and put a period after it. We will call this period a "decimal." The trick is to always make sure this decimal is directly above and below each other when you stack your numbers. If it isn't, then your answer will be wrong. Take a look at the picture below.


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You start by adding the first 2 numbers stacked on top of each other on the right that are closest to the decimals. If the answer is more than 9, write down the last half of your answer below those 2 numbers. Take the first half and put it on top of the numbers to the left. This is called "carrying." If your answer is 9 or less, you don't have to worry about carrying, you simply put the answer below the 2 numbers. When you finish adding the numbers closest to the decimals on the right, you go to the next numbers stacked on top of each other to the left and do the same thing again. You keep doing this, going to the left, until there aren't any more numbers to add. Take a look at the picture below, then I'll explain what happens in each step.


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In step 1 we stack the numbers up with the decimals on top of each other. We begin by adding the first 2 red numbers beside the decimals on the right. 2+9=11. The answer is 11. 11 is larger than 9 so we must carry.
In step 2 we put the last half of the answer under the 2 numbers that we added and "carry" the first half of the answer over and on top of the next 2 numbers to the left. The number that we carried is shown in green.
In step 3 we move over to the next 2 numbers to the left, which are shown in red. We have to add these 3 numbers now. 1+4+9=14. The answer is 14. The answer is larger than 9 this time also, so we must carry again.

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In step 4 we put the last half of the answer under the 3 numbers that we added and "carry" the first half of the answer over and on top of the next number to the left. The number that we carried is shown in green.
In step 5 there is no number on the bottom. All we need to do is add the top 2 numbers shown in red. 1+1=2. The answer is 2. 2 is smaller than 9, so we don't have to carry this time.
In step 6 we write our answer below the 2 numbers we added.
There are no more numbers to the left. We now have our final answer. We found that 142+99=241. Wasn't that easy! Look over things again and make sure you understand how to "carry" and stack the numbers correctly.


Subtraction is also a little trickier with larger numbers. Instead of carrying, with subtraction we sometimes have to "borrow" 1 from the top number to the left. We stack the numbers up just like we did with addition. When subtracting, you must always put the larger number on top. We start with the 2 numbers on the far right again and work our way to the left, just like we did with addition. We first check to see if the top number is larger than the bottom number. If it is, we just subtract and write the answer below the 2 numbers, then move to the next 2 numbers on the left.
If the top number is smaller than the bottom number, we have to "borrow" 1 from the top number on the left. When we borrow 1 from the top number to the left, we are actually subtracting 1 from that number. It also adds 10 to the top number of the 2 numbers we are working with. After we borrow we are able to subtract the bottom number from the top number, because it is now larger than the bottom number. We subtract and write our answer below the 2 numbers and move to the next 2 numbers on the left. We keep doing this until we run out of numbers to subtract. Take a look at the picture below, then I'll explain what happens in each step.


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In step 1 we stack the numbers up with the decimals on top of each other like we did with addition. We also make sure that we have the larger number on top. We begin with the first 2 red numbers beside the decimals on the right. We check to see if the top number is larger than the bottom number. In this problem the top number is smaller, so we will have to borrow 1 from the top number to the left.
In step 2 we borrow 1 from the top number on the left, which is 4. Since we are taking 1 away, we have to subtract 1 from 4, crossing out the 4 and writing the answer 3 on top, which is shown in green. This adds 10 to the top number we are working with, which is 1. We add 10 to 1, crossing out the 1 and writing the answer 11 on top, which is shown in red. Since the top number is now larger we can subtract. 11-9=2. We write the 2 down below the 2 numbers we are working with.
In step 3 we move over to the next 2 numbers to the left, which are shown in red. We check to see if the top number is larger than the bottom number. It isn't so we're going to have to borrow again.

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In step 4 we borrow 1 from the top number to the left, which is 1. We subtract 1 from 2, crossing out the 2 and writing the answer 1 on top, which is shown in green. This adds 10 to the top number we are working with, which is 3. We add 10 to 3, crossing out the 3 and writing the answer 13 on top, which is shown in red. 13 is larger than 9, so we can now subtract. 13-9=4. We write the 4 down below the 2 numbers we are working with.
In step 5 all we have left is the 1 by itself. You could think of it like there was an imaginary 0 below it and 1-0=1.
In step 6 we bring the 1 down and put it below. We now have our final answer! 241-99=142! That wasn't so bad was it? Look it over again. Make sure you understand how "borrowing" works. People often wonder why you add 10 when borrowing, when you only borrowed 1. Look carefully at the picture below and maybe it will help. The number to the left is always 10 more than the same number on the right. Think about it carefully.


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


*** Addition ***

2+3=5
(is the same as)
3+2=5

5+4+3=12
(is the same as)
4+3+5=12

*** Subtraction ***

3-1=2
(is NOT the same as)
1-3=??

10-5=5
(is NOT the same as)
5-10=??


You made it! We're through with the lesson! Now's a good time to scroll back up the page and make sure you understood 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 see where you might need to practice more. Good luck and thanks for reading through the lesson!


QUIZ

Question 1:

2 + 1 = _____

Answer:
A:  1
B:  2
C:  3
D:  4

Question 2:

4 + 2 = _____

Answer:
A:  2
B:  4
C:  6
D:  8

Question 3:

5 + 5 = _____

Answer:
A:  0
B:  1
C:  5
D:  10

Question 4:

2 + 3 + 4 = _____

Answer:
A:  5
B:  6
C:  7
D:  9

Question 5:

79 + 52 = _____

Answer:
A:  121
B:  131
C:  1211
D:  None of the above

Question 6:

2 - 1 = _____

Answer:
A:  0
B:  1
C:  2
D:  3

Question 7:

4 - 2 = _____

Answer:
A:  1
B:  2
C:  4
D:  6

Question 8:

5 - 5 = _____

Answer:
A:  0
B:  1
C:  2
D:  10

Question 9:

199 - 0 = _____

Answer:
A:  0
B:  198
C:  199
D:  None of the above

Question 10:

64 - 55 = _____

Answer:
A:  9
B:  10
C:  19
D:  None of the above




~ Other Links ~

Addition Surprise

High - Rise Math

Let's Do Math!

Coolmath4kids - Addition

Coolmath4kids - Subtraction

Harcourt E-Lab

Harcourt Math Glossary


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