Associative, Commutative and Distributive Laws

Wow! What a mouthful of words! But the idea is simple.

Commutative Laws

The "Commutative Laws" just mean that you can swap numbers over and still get the same answer when you add, or when you multiply.

a + b = b + a
a × b = b × a

Examples:

You can swap when you add:

3 + 6 = 6 + 3

You can swap when you multiply:

2 × 4 = 4 × 2

Associative Laws

The "Associative Laws" mean that it doesn't matter how you group the numbers (ie which you calculate first) when you add, or when you multiply.

(a + b) + c = a + (b + c)
(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 even 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.

It means you get the same answer when you:

add up some numbers then do a multiply, or
do each multiply separately then add them

Like this:

(a + b) × c = a × c + b × c

Examples:

This:	(2 + 4) × 5 = 6 × 5 = 30
Has the same answer as this:	2×5 + 4×5 = 10 + 20 = 30

This:	(6 - 4) × 3 = 2 × 3 = 6
Has the same answer as this:	6×3 - 4×3 = 18 - 12 = 6

Uses:

Sometimes it is easier to break up a difficult multiplication:

What is 204 × 6?
204 × 6 = 200×6 + 4×6 = 1,200 + 24 = 1,224

Or to combine:

What is 6 × 16 + 4 × 16?
6 × 16 + 4 × 16 = (6+4) × 16 = 10 × 16 = 160

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

Conclusion

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 × c + b × c