Download Associative Law

The Basic Operations and Special Properties
http://www.youtube.com/watch?v=MOYNs-FXKfY
In basic mathematics there are many ways of saying the same thing:
Symbol
+
Words Used
Addition, Add, Sum, Plus, Increase, Total
-
Subtraction, Subtract, Minus, Less, Difference,
Decrease, Take Away, Deduct
×
Multiplication, Multiply, Product, By, Times,
Lots Of
÷
Division, Divide, Quotient, Goes Into, How
Many Times
Addition is ...
... bringing two or more numbers (or things) together to make a new total.
The numbers to be added
together are called the
"Addends":
Subtraction is ...taking one number away from another.
Minuend - Subtrahend = Difference
Minuend: The number that is to be subtracted from.
Subtrahend: The number that is to be subtracted.
Difference: The result of subtracting one number from another.
Multiplication is ...
.. (in its simplest form) repeated addition.
Here we see that 6+6+6 (three 6s)
make 18
It could also be said that
3+3+3+3+3+3 (six 3s) make 18
Division is ...
... splitting into equal parts or groups. It is the result of "fair sharing".
Division has its own special words to remember.
Let's take the simple problem of dividing 22 by 5. The answer is 4, with 2 left over.
Here we illustrate the important words:
Which is the same as:
Average
You calculate the average by adding up all the values, then divide by how
many values.
Example: What is the average of 9, 2, 12 and 5?
Add up all the values: 9 + 2 + 12 + 5 = 28
Divide by how many values (there are four of them): 28 ÷ 4 = 7
So the average is 7
This makes it the Additive Identity, which is just a special way of saying "add 0
and you get the identical number you started with".
Special Properties
http://www.youtube.com/watch?v=c3Z59HjNxEQ
Additive Identity
Adding zero to a number leaves it unchanged:
a+0=0+a=a
Additive Inverse
What you add to a number to get zero.
The negative of a number.
Example:
The additive inverse of -5 is 5, because -5 + 5 = 0.
The additive inverse of +5 is -5 as well.
Multiplicative Inverse
Another name for Reciprocal.
When you multiply a number by its "Multiplicative Inverse" you get 1.
Example: 8 × (1/8) = 1
Associative Law
The "Associative Laws" say:
* It doesn't matter how you group the numbers when you add.
* It doesn't matter how you group the numbers when you multiply.
(In other words it doesn't matter which you calculate first.)
Example addition: (6 + 3) + 4 = 6 + (3 + 4) Because 9 + 4 = 6 + 7 = 13
Example multiplication: (2 × 4) × 3 = 2 × (4 × 3) ' cause 8 × 3 = 2 × 12 = 24
Commutative, Associative and Distributive Laws
Wow! What a mouthful of words! But the ideas are simple.
The "Commutative Laws" say you can swap numbers over and still get the same
answer .... when you add:
a+b = b+a
Example:
... or when you multiply:
a×b = b×a
Example:
Associative Laws
The "Associative Laws" say that it doesn't matter how you group the numbers (i.e.
which you calculate first) ...
... when you add:
(a + b) + c = a + (b + c)
... or when you multiply:
(a × b) × c = a × (b × c)
Examples:
This:
(2 + 4) + 5 = 6 + 5 = 11
Has the same answer as this:
2 + (4 + 5) = 2 + 9 = 11
This:
(3 × 4) × 5 = 12 × 5 = 60
Has the same answer as this:
3 × (4 × 5) = 3 × 20 = 60
Uses: Sometimes it is easier to add or multiply in a different order:
What is 19 + 36 + 4?
19 + 36 + 4 = 19 + (36 + 4) = 19 + 40 = 59
Or to rearrange a little:
What is 2 × 16 × 5?
2 × 16 × 5 = (2 × 5) × 16 = 10 × 16 = 160
Distributive Law
The "Distributive Law" is the BEST one of all, but needs careful attention.
This is what it lets you do:
3 lots of (2+4) is the same as 3 lots of 2 plus 3 lots of 4
So, the 3× can be "distributed" across the 2+4, into 3×2 and 3×4
And we write it like this:
a × (b + c) = a × b + a × c
Try the calculations yourself:

3 × (2 + 4) = 3 × 6 = 18

3×2 + 3×4 = 6 + 12 = 18
Either way gets the same answer.
In English we can say:
You get the same answer when you:


multiply a number by a group of numbers added together, or
do each multiply separately then add them
Uses:
Sometimes it is easier to break up a difficult multiplication:
Example: What is 6 × 204 ?
6 × 204 = 6×200 + 6×4 = 1,200 + 24 = 1,224
Or to combine:
Example: What is 16 × 6 + 16 × 4?
16 × 6 + 16 × 4 = 16 × (6+4) = 16 × 10 = 160
You can use it in subtraction too:
Example: 26×3 - 24×3
26×3 - 24×3 = (26 - 24) × 3 = 2 × 3 = 6
You could use it for a long list of additions, too:
Example: 6×7 + 2×7 + 3×7 + 5×7 + 4×7
6×7 + 2×7 + 3×7 + 5×7 + 4×7 = (6+2+3+5+4) × 7 = 20 × 7 = 140
And those are the Laws!
But Not ...
But don't go too far!
The Commutative Law does not work for division:
Example:


12 / 3 = 4, but
3 / 12 = ¼
The Associative Law does not work for subtraction:
Example:


(9 – 4) – 3 = 5 – 3 = 2, but
9 – (4 – 3) = 9 – 1 = 8
The Distributive Law does not work for division:
Example:


24 / (4 + 8) = 24 / 12 = 2, but
24 / 4 + 24 / 8 = 6 + 3 = 9
Summary
Commutative Laws:
a+b = b+a
a×b = b×a
Associative Laws:
(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)
Distributive Law:
a × (b + c) = a × b + a × c
Expanded Notation
Writing a number to show the value of each digit.
It is shown as a sum of each digit multiplied by its matching place value
(units, tens, hundreds, etc.)
For example: 4,265 = 4 x 1,000 + 2 x 100 + 6 x 10 + 5 x 1
Here are some of zero's properties:
Property
Example
a+0=a
4+0=4
a−0=a
4−0=4
a×0=0
6×0=0
0/a=0
0/3 = 0
a / 0 = undefined (dividing by zero is undefined)
7/0 = undefined
0a = 0 (a is positive)
04 = 0
00 = indeterminate
00 = indeterminate
0a = undefined (a is negative)
0-2 = undefined
0! = 1 ("!" is the factorial function)
0! = 1