MULTIPLICATION

With the trend towards more and earlier calculator usage,
some people have lost a comfort with basic arithmetic operations like [beautiful math coming... please be patient] $\;5\cdot 7 = 35\;$ and [beautiful math coming... please be patient] $\;8 + 6 = 14\;$.
It is a waste of valuable time to use your calculator for problems such as these.
In this section, your basic multiplication skills are brought ‘up to speed’
so you won't be wasting mental energy on arithmetic and will be able to concentrate on higher-level ideas.

In mathematics, a vertically centered dot ‘$\;\cdot\;$’ is often used to denote multiplication.
Thus, [beautiful math coming... please be patient] $\;3\cdot 5\;$ denotes ‘$\,3\,$ times $\,5\,$’.

Consider the multiplication problem [beautiful math coming... please be patient] $\;3\cdot 5\;$:

One interpretation is $3\,$ groups of $\,5\;$, so: $3\cdot 5 = 5 + 5 + 5 = 15$
Another interpretation is $5\,$ groups of $\,3\;$, so: $3\cdot 5 = 3 + 3 + 3 + 3 + 3 = 15$

Thus, multiplication is a shorthand for repeated addition.

Algebra uses letters to represent numbers. (LOTS more on this later on!)
The expression [beautiful math coming... please be patient] $\;2x\;$ means ‘$\,2\,$ times $\,x\,$’. This is a shorthand for the addition problem $\;x + x\;$.
Similarly, the expression [beautiful math coming... please be patient] $\;3y\;$ means ‘$\,3\,$ times $\,y\,$’. This is a shorthand for the addition problem [beautiful math coming... please be patient] $\;y + y + y\;$.

Thus, for example, [beautiful math coming... please be patient] $\;5\cdot 3x\;$ means [beautiful math coming... please be patient] $\;3x + 3x + 3x + 3x + 3x\;$, which is $\;15x\;$.
It's much easier to think in terms of multiplication rather than repeated addition, so you immediately get [beautiful math coming... please be patient] $\;5 \cdot 3x = 15x\;$.
Similarly, [beautiful math coming... please be patient] $\;6\cdot 7t = 42t\;$.

To commute means to change places.

The Commutative Property of Multiplication states that for all numbers [beautiful math coming... please be patient] $\;x\;$ and [beautiful math coming... please be patient] $\;y\;$,   [beautiful math coming... please be patient] $x \cdot y = y\cdot x\;$.
That is, you can change the places of the numbers in a multiplication problem, and this does not affect the result.

The expression [beautiful math coming... please be patient] $\;x\cdot y\;$ can more simply be written as [beautiful math coming... please be patient] $\;xy\;$.
That is, juxtaposition (placing two things side-by-side) can be used to denote multiplication.
Of course, this notation can't be used with numbers like $\,2\,$ and $\,3\,$,
because ‘$\,23\,$’ means the number twenty-three, not $\,2\,$ times $\,3\,$!

So, we can restate the Commutative Property of Multiplication:   For all numbers [beautiful math coming... please be patient] $\;x\;$ and [beautiful math coming... please be patient] $\;y\;$,   [beautiful math coming... please be patient] $xy = yx\;$.

If you're a sociable person, then you probably like being in groups;
i.e., you like to associate with other people.
In mathematics, associative laws have to do with grouping.

The Associative Property of Multiplication states that for all numbers [beautiful math coming... please be patient] $\,x\,$, $\,y\,$, and $\,z\,$,   [beautiful math coming... please be patient] $(x\cdot y)\cdot z = x\cdot(y\cdot z)\;$.
Notice that the order in which the numbers are listed on both sides of the equation is exactly the same; only the grouping has changed.
The Associative Property of Multiplication states that in a multiplication problem, the grouping of the numbers does not affect the result.

Thanks to the associative property, we can write things like [beautiful math coming... please be patient] $\;2\cdot 3\cdot 4\;$ without ambiguity!
Think about this—if the grouping mattered, then [beautiful math coming... please be patient] $\;(2\cdot 3)\cdot 4\;$ and [beautiful math coming... please be patient] $\;2\cdot (3\cdot 4)\;$ would give different results,
so you'd always have to use parentheses to specify which way it should be done.

Multiplying a number by [beautiful math coming... please be patient] $\;1\;$ does not change it.
In other words, multiplying by [beautiful math coming... please be patient] $\;1\;$ preserves the identity of the original number.
More precisely:   for all numbers [beautiful math coming... please be patient] $\,x\,$,   [beautiful math coming... please be patient] $x\cdot 1 = 1\cdot x = x\;$.
For this reason, the number [beautiful math coming... please be patient] $\;1\;$ is called the multiplicative identity.

Multiplying a number by [beautiful math coming... please be patient] $\;0\;$ always gives [beautiful math coming... please be patient] $\;0\;$.
More precisely:   for all numbers [beautiful math coming... please be patient] $\,x\,$,   [beautiful math coming... please be patient] $x\cdot 0 = 0\cdot x = 0\;$.
This fact is sometimes called the Multiplication Property of Zero.

Master the ideas from this section
by practicing both exercises at the bottom of this page.

When you're done practicing, move on to:
Divisibility

 
 

In this exercise, you will practice multiplication problems of the form [beautiful math coming... please be patient] $\;x \cdot y\;$,
where $\,x\,$ and $\,y\,$ can be any of these numbers: $\;0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $11$, and $12\,$.

If desired, check the multiplication table(s) you want to practice:
0 1 2 3 4 5 6 7 8 9 10 11 12 no variables in problems    

    
(an even number, please)
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
AVAILABLE MASTERED IN PROGRESS

(MAX is 18; there are 18 different problem types.)