Download For example: 3(x+5) + 2(x-4) can not be simplified since I don`t

Explanation phase
Try this. To distribute something means to hand it out. If you distribute a test paper to
your class, you give a test to each person in the class.
The distributive property usually means that multiplication (*) distributes over addition
(+). It is stated in the form of the following formula:
a*(b+c) = a*b + a*c,
where a, b, and c can be any numbers.
In other words, you can either add b and c together first or then multiply by a, or you can
multiply a by b and a by c, and add those products, and the result will be the same in
either case.
Let's take a specific case: 5 x (2 + 3). The distributive property says that this is the same
as (5 x 2) + (5 x 3). When we write:
5 x (2 + 3) = (5 x 2) + (5 x 3)
we say that we have distributed the five to the two and the three. Both sides of the
equation are equal so they will work to the same answer:
5 x (2 + 3)
=5x5
= 25
(5 x 2) + (5 x 3)
= 10 + 15
= 25
Using our example, the Distributive Property says that five times the sum of two plus
three is equal to the product of five times two plus the product of five times three.
Using coins to show this, take twenty-five pennies. First, make five stacks of two pennies
and five stacks of three pennies. This represents the right-hand side of the equation
above: five times two plus five times three (or written mathematically: (5 x 2) + (5 x 3)).
Now pair each stack of two pennies with a stack of three pennies. This represents the left
side of the example above: five groups, two plus three in each group (or written
mathematically: 5 x (2 + 3)).
What a strange world it would be if you could take twenty-five pennies and have more or
less than 25 based on how you grouped them!
For example, suppose the letter x stands for some (unknown) number, and 2x means
twice that number. Then if we know that
x + 2x = 15
then we know that x is the same as 1x (1 times x), and we can use the distributive
property to say that
x + 2x = 1x + 2x = (1+2)x = 3x
So we can rewrite our equation as
3x = 15
But now it's easy to find what x has to be; the number you can multiply by 3 to get 15 is
15 divided by 3, or 5.
The distributive property is really just common sense. My equation might mean
I bought one CD for me and two for my big sister. The total cost was $15. I forget how
much a CD costs. Can you tell me?
Well, one CD and two CDs is the same as 3 CDs; so 1 times the cost of a CD plus 2 times
the cost of a CD is 3 times the cost of a CD. What I did to the equation is the same
thinking without the words.
Suppose you have some cookies, and you arrange them in a 12 by 2 rectangle:
@@@@@@@@@@@@
@@@@@@@@@@@@
And suppose you have some other cookies, and you arrange them in a 12 by 3 rectangle:
************
************
************
If you put these two rectangles together, you get this:
@@***
@@***
@@***
@@***
@ @ * * * = (12 * 2) + (12 * 3)
@@***
@@***
@@***
@@***
@@***
@@***
@@***
@@***
The number of items in each rectangle is the height times the width. This includes the
larger rectangle. So we have two different ways that we can compute the number of
cookies without counting them all:
@@***
@@***
@@***
@@***
@@***
@@***
@@***
@@***
@@***
@@***
@@***
@@***
@@***
= (12 * 2) + (12 * 3)
= 12(2 + 3)
Sometimes using the property to rewrite an expression makes things a lot easier to
compute. For example, which of these would you rather compute?
1. (210 * 196) + (210 * 54) = ?
2. 210(196 + 54) = 210 * 250 = ?
Sometimes you can use it to turn one hard operation into two easy ones:
210 * 196 = 210(200 - 4)
= 210*200 - 210*4
= 42000 – 840
A different problem would be
(49)70 = (40 + 9)70
= (40)70 + (9)70
= 2800 + 630
= 3430
Actually, this is a good trick to use to do multiplication in your head, so it's worth
spending a little time practicing it.
But where the distributive property really becomes important is in algebra, where you'll
be using it about every 30 seconds to simplify expressions like
________________________
|
|
|
v
-------------------3(x + 2y) + 5(2x + 4y) = (3x + 6y) + (10x + 20y)
----------------|
^
|________________________|
= 3x + 10x + 6y + 20y
--------- ---------|
|
v
v
--------- --------= x(3 + 10) + y(6 + 20)
= 13x + 26y
--------|
v
--------= 13(x * 2y)
Each one of the arrows represents an application of the distributive property.
That may not make sense, but let me give you a concrete example. I will show you that
division (/) does NOT distribute over addition (+).
In other words,
a/(b+c) is not equal to a/b + a/c.
To check, just plug in some numbers:
5/(7+3) is not equal to 5/7 + 5/3, since
5/(7+3) = 5/10 = 1/2,
but 5/7 + 5/3 = 15/21 + 35/21 = 50/21, which is not equal to 1/2.
I would say that the 'best' thing to do would be to make sure that _every_ student in the
class understands the distributive property; and I mean _really_ understands it: can
illustrate it with diagrams, can apply it to long and short sequences of terms involving
both positive and negative numbers, sees why it's the basis for adding fractions, i.e.,
2 3 1 1
1 2+3
- + - = 2*- + 3*- = (2+3)*- = --7 7 7 7
7 7
and so on.
Once you've done that, you can simply point out that
-(b - a) = -1(b - a)
= -1*b - -1*a
= -b + a
=a-b
In fact, once your students really understand the distributive property, you don't really
have to 'teach' this to them at all.
So that's 'best'. What about 'simplest'? The simplest way I can think of teach this is to
note that we can use arrows to represent subtraction:
------|----------|-----a
b
----------->
b-a
Here, (b-a) is whatever we have to add to a to get to b. It's simple enough to show this
symbolically:
a + (b-a) = a + b - a
=b+a-a
=b
Once you've accepted this, then you can extend the drawing like this:
a-b
<----------------|----------|-----a
b
----------->
b-a
which shows that (a-b) and (b-a) are the same size, but differ in sign.