**Concept 1**The counting numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc.

**Concept 2**The Adding Chart

**Concept 3**The addition patterns

The addition patterns:

(first number) + (second number) = sum

(second number) + (first number) = sum

(first number) + 0 = (first number)

(second number) + 0 = (second number)

(first number) + (second number) = (second number) + (first number)

To put it in a more concise way, let "A" stand for the first number and "B" stand for the second number.

A + B = sum

B + A = sum

A + 0 = A

B + 0 = B

A + B = B + A

The last statement is known as the

**Law of Commutativity**

Addition is known as a commutative operation because it makes no difference which order the numbers are added in.

For example:

A + B + C = A + C + B = B + C + A = B + A + C = C + B + A = C + A + B

We shall go on to learn subtraction. Subtraction is adding in reverse. In former lessons you had to solve problems like:

1 + 1 = ?

3 + 4 = ?

5 + 2 = ?

9 + 6 = ?

2 + 5 = ?

In the previous problems, you were given the first number and the second number and had to compute the sum. That is, you were given the addends and had to compute the sum. In the next few problems we will work with problems where the sum is given but one of the addends is missing. We must compute the addends.

Let's take the problem: 3 + ? = 7

The problem is asking: What, when added to 3, gives us the sum of 7?

Taking a quick look at the addition chart we will find that 3 + 4 = 7. Therefore, when 4 is added to 3, it gives us 7. How about this one:

? + 5 = 8

What, when added to 5 gives us the sum of 8?

Again, a quick look at the adding chart should give us the answer: 3 + 5 = 8.

There is a more direct way to solve these problems without the use of an adding chart. The process is called

**subtraction**.

Instead of doing 3 + ? = 7, we subtract and thus do

7 - 3 = ?

Doing this visually, let's take seven objects

7 objects:

(object #1)

(object #2)

(object #3)

(object #4)

(object #5)

(object #6)

(object #7)

And now delete three of the seven objects. (Any three, it makes little difference.)

Delete 3 objects:

(object #1)

(object #2)

(object #3)

(object #4)

(object #6)

(object #7)

Objects remaining:

(object #1)

(object #2)

(object #3)

(object #4)

Now count the remaining objects: 1... 2... 3... 4.

We have four remaining. Thus

7 - 3 = 4

Let's try another example:

8 - 6 = ?

8 objects:

(object #1)

(object #2)

(object #3)

(object #4)

(object #5)

(object #6)

(object #7)

(object #8)

Delete 6 objects:

(object #2)

(object #3)

(object #4)

(object #6)

(object #7)

Objects remaining:

(object #4)

(object #7)

Count the remaining objects: 1... 2.

Thus 8 - 6 = 2.

The number 8 is known as the

**minuend**of the equation; the number 6 is known as the

**subtrahend**of the equation; the number 2 is known is the

**difference**of the equation. 2 is the difference of and 8 and 6.

Another one: 5 - 5 = ?

5 objects:

(object #1)

(object #2)

(object #3)

(object #4)

(object #5)

Delete 5 objects:

(object #2)

(object #3)

(object #4)

(object #5)

Objects remaining:

(none)

Count the remaining objects: (none)

Thus 5 - 5 = 0.

Try 3 - 0 = ?

3 objects:

(object #1)

(object #2)

(object #3)

Delete 0 objects:

(object #1)

(object #2)

(object #3)

Objects remaining:

(object #1)

(object #2)

(object #3)

Thus 3 - 0 = 3.

Try 3 - 5 = ?

**This cannot be done**(unless we invent a new type of number). We are asking to remove five objects from a group that does not contain that many objects. Five is a bigger number than three. Five is greater than three. Mathematically written, it is

5 > 3

The concept of "greater than" can wait for another lesson. For now, let's focus on some properties of subtraction.

General formula: (Minuend) - (Subtrahend) = (Difference)

General pattern: (number) - 0 = (number)

3 - 5 is not the same as 5 - 3

5 - 3 = 2 while 3 - 5 cannot be done in our elementary system of arithmetic.

Subtraction property 1: A - B is not equal to B - A.

Subtraction property 2: A - 0 = A

Subtraction property 3: A - A = 0

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