Addition is when you add to something. When you add numbers, you keep moving right on the number line. When you add similar items, you can simply count the total. For example, if you have 3 oranges and you add 2 oranges, you will have 5 oranges. When we are just dealing with numbers, the objects happen to be ones (1's), so if we add 5 to 3, we have 5 ones and 3 ones, which makes 8 ones or simply 8.
5 + 3 = ( 1+1+1+1+1 ) + ( 1+1+1 ) = 8
Subtraction is when you take away. When you subtract numbers, you keep moving left on the number line. When you take away similar items, you count what is left. For example, if you have 5 oranges and you take away 2 oranges, you will have 3 oranges left. When we are just dealing with numbers, if we subtract 2 from 5, we have 5 ones and take away 2 ones, leaving us with 3 ones or simply 3.
5 - 2 = ( 1+1+1+1+1 ) - ( 1+1 ) = 1-1 + 1-1 +1 +1 +1 = 0 + 0 + 1 + 1 + 1 = 3
Multiplication is when you are dealing in groups. Multiplication can be thought of as repeated addition. For example, 2 × 3 = 2 groups of 3 = ( 1+1+1 ) + ( 1+1+1 ) = 6. Note that in the given example
2 × 3 = 3 × 2 = 3 groups of 2 = ( 1+1 ) + ( 1+1 ) + ( 1+1 ) = 6 (commutative property). Of course, you don't want to reduce every multiplication problem down to addition, so it is important that you memorize the following multiplication table:
Take a number on any vertical orange square, multiply it times a number on any horizontal orange square and the answer will appear where the respective horizontal and vertical columns meet. For example, 6 × 2 = 12.
Note how each column or row is incremented by the number on the orange border; i.e. each number in the 3 row is 3 larger than the number preceding it.
Also note that the blue squares are all products of the same number; i.e. 1 = 1 × 1, 4 = 2 × 2, 9 = 3 × 3, 16 = 4 × 4, etc. These are called perfect squares.
Division is when you are splitting something into equal size groups. It is the inverse or opposite of multiplication. For example, when multiplying, you might be asked 'How many candy do you have if you have 5 bags with 3 candy in each bag?' In division you might be asked a very similar question, 'How many candy can you put in 5 bags if you have a total of 15 candy and you must put the same number of candy in each bag?'
The first question involves multiplication: 5 × 3 = 15.
The second question involves division: 15 ÷ 5 = 3.
Look at multiplication table above, if you are given a number on a vertical orange square and a product (not orange) that is on the same line, then the answer will be the number that is in the same column as the product and on a horizontal orange square. For example; to find how many groups of 9 can be made from 54 items, find the vertical orange square with the number 9, move left on that line until you find the number 54 and finally move straight up from 54 until you get to the row of horizontal orange squares and you will find the answer, 6.