What's the difference between Distributive and Commutative properties, and how can I use them?

Question

 I'm learning about Distributive and Commutative properties, and I don't really understand it. I have a math test I have to take on it, but I'm afraid I won't be able to pass with my current knowledge. Can someone please help me understand it, even if its just by a little bit?

Answer 1

2(a+3b)

 

The distributive property applies here.  You can't combine the a or the b with anything else until you get them outside the parentheses.  You have to "distribute" the multiplication by two across the a and b.

You get 2a + 6b.

I like to think of parentheses as a box.  The label on this box says there's one a and 3 b's inside, and there are 2 boxes. When you open the boxes and take everything  out, you have 2 a's and 6 b's.

 

If there is a subtraction inside the parentheses, the rule is the same.  It just isn't as easy to illustrate.

2(a-3b) = 2a - 6b

 

a+b+c = a + c + b = c + a + b

This is an example of the commutative property.  You can  change ("commute") the order in which you add things together.

if you have a quarter a dime and a nickel, you can start with 25, add 10 to get 30, then add 5 to get 40.

Or if you want, you can start with the nickel, then the dime and finally the nickel. You get 5, then 15, then 40.

The order of addition doesn't matter as long as everybody gets added in.

 

The commutative property also works for multiplication.

2 × 3 × 4 = 24

That's 2 groups of 3 = 6, then 4 groups of 6 = 24.

3 × 4 × 2 = 24

That's 3 groups of 4 = 12, then 2 groups of 12 = 24.

Again, the order of multiplication doesn't matter.

 

Does this help?