Multiplication is “repeated addition.”
To explore further logic that follows from the idea of multiplication as “repeated addition,” go to the link
MATH MILESTONES #A4, MULTIPLICATION
Many a time the problem with multiplication gets resolved by getting the student to count on the ten fingers “by two,” “by three,” “by four,” etc. When the student is counting on his fingers, say, “by three,” each finger has a value of three. As he counts “by three,” he adds three at each count. He may then write down the count in a column. This will then be the multiplication table for three.
By repeated addition, one may easily produce multiplication tables, and then use them to solve problems.
The proper approach is to create the multiplication table, as above, many times using “repeated addition.” Knowing the techniques of addition from MS 02: ADDITION can be of great help. You are building the skill in multiplication as an extension of the skill in addition. This approach is better than simple memorization.
The real shortcuts in math come from thinking with the basics, such as,

A number multiplied by zero = the number “added repeatedly” zero times = zero

A number multiplied by one = the number “added repeatedly” once = the number

A number multiplied by 10 = the digits shift one place value to the left = the number with a zero attached (e.g., 3 x 10 = 30; 12 x 10 = 120).
Check out the above with repeated addition on abacus per MILESTONE 1: Numbers & Place Values]
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The following is the most amazing property of repeated addition.
4 x 17 = 4 added repeatedly 17 times
= 4 added repeatedly 10 times + 4 added repeatedly 7 times
This may be written as,
4 x 17 = 4 x (10 + 7) = (4 x 10) + (4 x 7)
One may now compute this mentally as 40 +28 = 68. Here, the key property is
4 x (10 + 7) = (4 x 10) + (4 x 7)
This ia also written in its general form as
a x (b + c) = (a x b) + (a x c)
And we recognize it as the Distributive Property.
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Since one operation of multiplication contains many opearation of addition, multiplication takes precedence over addition in mixed operations. For example,
3 + 2 x 4 = 11 (not 20)
Here are some videos from the Khan Academy that explain multiplication:
Multiplication 2: The Multiplication Tables
Multiplication 3: 10,11,12 times tables
Multiplication 4: 2digit times 1digit number
Multiplication 5: 2digit times a 2digit number
Multiplication 6: Multiple Digit Numbers
Multiplication 7: Old video giving more examples
Mulitplication 8: Multiplying decimals (Old video)
Why Lattice Multiplication Works
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