MULTIPLICATION

LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
 

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, $\;5\cdot 3x\;$ means $\;3x + 3x + 3x + 3x + 3x\;$, which is $\;15x\;$.
It's much easier to think in terms of multiplication rather than repeated addition, so you immediately get $\;5 \cdot 3x = 15x\;$.
Similarly, $\;6\cdot 7t = 42t\;$.

To commute means to change places.

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

The expression $\;x\cdot y\;$ can more simply be written as $\;xy\;$.
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 $\;x\;$ and $\;y\;$,   $xy = yx\;$.

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 $\,x\,$, $\,y\,$, and $\,z\,$,   $(x\cdot y)\cdot z = x\cdot(y\cdot z)\;$ .
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 $\;2\cdot 3\cdot 4\;$ without ambiguity!
Think about this—if the grouping mattered, then $\;(2\cdot 3)\cdot 4\;$ and $\;2\cdot (3\cdot 4)\;$ would give different results,
so you'd always have to use parentheses to specify which way it should be done.

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

Multiplying a number by $\;0\;$ always gives $\;0\;$.
More precisely:   for all numbers $\,x\,$,   $x\cdot 0 = 0\cdot x = 0\;$.
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.)