Properties of Addition and Multiplication

Finn reads this

Math has four operations, but only two of them follow every rule: addition and multiplication. The rules are called properties. Each one is named after an everyday word, so if you know the word, you know the rule.

Commutative

Finn reads this

Root word: commute. From Latin com- (together) and mutare (to change), the same root as mutate. To commute is to travel back and forth, trading places.

What it lets you do: swap the order.

a + b = b + aa × b = b × a

x + 5 = 5 + x4 × 7 = 7 × 4

ElleMath-speak

By the commutative property of addition, a + b = b + a.

FinnPlain English

When you add or multiply, you can swap the order.

Associative

Finn reads this

Root word: associate. From Latin ad- (to) and socius (companion), the same root as social. An associate is a partner. Parentheses show who is partnered up.

What it lets you do: change partners.

(a + b) + c = a + (b + c)(a × b) × c = a × (b × c)

(17 + 6) + 4 = 17 + (6 + 4) = 17 + 10 = 27

ElleMath-speak

By the associative property, the grouping can change.

FinnPlain English

When you add or multiply, it doesn't matter who pairs up first.

Distributive

Finn reads this

Root word: distribute. From Latin dis- (apart) and tribuere (to give), the same root as contribute. To distribute is to hand something out to everyone.

The cookie story: Ana, Ben and Cal are a group. "Distribute 2 cookies to the group" means every kid gets 2. Nobody gets skipped.

2(a + b + c) = 2a + 2b + 2c

Finn reads this

Uh oh. The kids were misbehaving. Undistribute the cookies! Take 2 back from every kid.

2a + 2b + 2c = 2(a + b + c)

You can only take 2 back if every kid has 2 to give. The math-speak word for undistributing is factoring out.

ElleMath-speak

By the distributive property, a(b + c) = ab + ac.

FinnPlain English

Hand the outside number to every term inside the parentheses.

The helpers: identity and inverse

Finn reads this

Identity, from Latin idem, "the same". Adding 0 or multiplying by 1 keeps a number the same: x + 0 = x, and 1 × x = x. That is why x secretly means 1x.

Inverse, from invert, "to turn around". The opposite undoes adding: 5 + ⁻5 = 0. The reciprocal undoes multiplying: 34 × 43 = 1.

The catch

Elle reads this

Try the rules on the other two operations, subtraction and division.

9 − 4 = 5, but 4 − 9 = ⁻58 ÷ 2 = 4, but 2 ÷ 8 = 14

Subtraction and division break the rules. More about this on the next tile in this lesson.

Let me try a few

NC standards: NC.7.EE.1.

Where to go next

Let me try a few5 problems on properties of addition and multiplication. Next: Tile 2, What If There's Subtraction or Division?Change them into addition and multiplication first All tilesAll 3 parts of this lesson. Tell us what you thought about this tileA few quick questions about just this tile. It takes a few minutes, and opens in a new tab so you keep your place.