## Step 1 Review

Check out this Video about Multiplication, Commutativity and Distribution properties.

From Purple Math:

**Distributive Property**

The Distributive Property is easy to remember, if you recall that "multiplication *distributes* over addition". Formally, they write this property as "*a*(*b* + *c*) = *ab* + *ac*". In numbers, this means, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses *(or factor something out)*; any time a computation depends on multiplying through a parentheses *(or factoring something out)*, they want you to say that the computation used the Distributive Property.

**Why is the following true? 2(***x*+*y*) = 2*x*+ 2*y*

Since they distributed through the parentheses, this is true **by the Distributive Property.**

**Use the Distributive Property to rearrange: 4***x*– 8

The Distributive Property either takes something through a parentheses or else factors something out. Since there aren't any parentheses to go into, you must need to factor out of. Then the answer is "**By the Distributive Property, 4 x – 8 = 4(x – 2)"**

"But wait!" you say. "The Distributive Property says multiplication distributes over *addition,* not *subtraction!* What gives?" You make a good point. This is one of those times when it's best to be flexible. You can either view the contents of the parentheses as the subtraction of a positive number ("*x* – 2") or else as the addition of a negative number ("*x* + (–2)"). In the latter case, it's easy to see that the Distributive Property applies, because you're still adding; you're just adding a negative.

The other two properties come in two versions each: one for addition and the other for multiplication. (Note that the Distributive Property refers to both addition and multiplication, too, but to both within just one rule.)

**Associative Property**

The word "associative" comes from "associate" or "group";the Associative Property is the rule that refers to grouping. For addition, the rule is "*a* + (*b* + *c*) = (*a* + *b*) + *c*"; in numbers, this means

2 + (3 + 4) = (2 + 3) + 4. For multiplication, the rule is "*a*(*bc*) = (*ab*)*c*"; in numbers, this means 2(3×4) = (2×3)4. Any time they refer to the Associative Property, they want you to regroup things; any time a computation depends on things being regrouped, they want you to say that the computation uses the Associative Property. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

**Rearrange, using the Associative Property: 2(3***x*)

They want you to regroup things, not simplify things. In other words, they do not want you to say "6*x*". They want to see the following regrouping: **(2×3) x**

**Simplify 2(3***x*), and justify your steps.

In this case, they *do* want you to simplify, but you have to tell why it's okay to do... just exactly what you've *always* done. Here's how this works:

2(3x) |
original (given) statement |

(2×3)x |
by the Associative Property |

6x |
simplification (2×3 = 6) |

**Why is it true that 2(3***x*) = (2×3)*x*?

Since all they did was regroup things, this is true **by the Associative Property.**

**Commutative Property**

The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "*a* +* b* = *b* + *a*"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "*ab *=* ba*"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.

**Use the Commutative Property to restate "3×4×***x*" in at least two ways.

They want you to move stuff around, not simplify. In other words, the answer is not "12*x*"; the answer is any two of the following:

**4 × 3 × x, 4 × x × 3, 3 × x × 4, x × 3 × 4, and x × 4 × 3**

**Why is it true that 3(4***x*) = (4*x*)(3)?

Since all they did was move stuff around (they didn't regroup), this is true **by the Commutative Property.**

Your brain likes repitition to move information into long term memory.

Watch, Review and Look Over your Math Notes from our whole group math lesson.

factor: a number that is multiplied by another number to get a product

product: the answer when 2 factors are multiplied