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## A Parent’s Guide To Algebra’s Basic Concepts – Properties Made Easy – Associative Property

Remember that the “Manipulation” properties of algebra (my name for them) provide the rules for working with (manipulating) numbers and/or terms and allow us *exchange* the order of operations. The first property we discussed was the commutative property of addition/multiplication. It allows us to change the *order* of numbers when adding or multiplying. In this article, we discuss the second of the “Manipulation” properties.

The second of the “Manipulation” properties: **Associative property for addition/multiplication**

In symbols, this property says: a + (b + c) = (a + b) + c (addition) or (bxc) = (axb) xc (multiplication) Remember that the x used here represents multiplication, not a variable

Like the commutative property, this property is NOT true for subtraction or division.

With numbers, the associative property looks like: 3 + (17 + 12) = (3 + 17) + 12 and 6 x ( 5 x 9) = (6 x 5) x 9

So how is this different from the commutative property? With the associative property, the order of the numbers does NOT change, we just change the* grouping*. We will move the ( ). Because? Again, this is about changing the order of operations ** Yes there is a reason to do it**. For example, in 3 + (17 + 19), the correct work order would be to do the first ( ). But I need my calculator for (17 + 19). this

*I should not*they need a calculator, but many students rely on the calculator and pick it up too quickly. If they just took a second to really look, they would see that applying the associative property changes the problem to (3 + 17) + 19. No calculator is needed here because 3 + 17 = 20 and 20 + 19 is 39. Similarly, 6 x (5 x 9) becomes (6 x 5) x 9 or 30 x 9 and 30 x 9 is an easier problem than 6 x 45. I can do 30 x 9 = 270 in my head. Anytime we can eliminate the use of the calculator, we’ve saved time.

As before, to use the associative property, all operations must be equal: all additions or all multiplications. And as before, if the operations are mixed, you need to use PEMDAS.

Stop crying! I hear you complain! – Do I have to remember the names? “How do I remember which property does what?” Of course, I can’t really hear __you__ whining; but this is traditionally where algebra students begin to grumble with exactly these questions. And the answers are…

Yes, you have to learn the names and also spell it correctly. Yes, I took spelling tests in Algebra class. You should have heard the commotion *this* caused! “This is not an English class.” Why do we have to write in math class?” It must be kind of funny, except for the fact that they really didn’t think they needed to know how to write any words that aren’t used in English class. I think the correct spelling is important in all classes. I know that being able to spell a word correctly helps with pronunciation and vice versa. Many students pronounce commutative as if it were communicative. There is NO ‘n’ in commutative and the root word is commute, not common.So spelling and practicing speaking out loud is beneficial.

The reason it’s important to know the right names is the same reason we all have names. It is much faster to know the names than to have to describe everything. It’s like asking about your son’s friend, Joey, instead of having to describe the little boy with freckles and red hair who lives three streets away. Names save a lot of time.

As for how to remember which property does what… I’ll give you two ways to remember each. Then choose the method that makes the most sense for you.

**1st method: By root word.**

The root word for commutative is to commute. As in commuting to school or work: Colorado Springs to Denver in the morning and Denver to Colorado Springs in the evening. The **order** of the cities is different, but the distance is 60 miles either way.

The root word for associative is associate. Who do you associate with? Who is in your **group**? The ( ) in these problems changes which numbers are grouped or matched first.

**2nd method: Use the first letters of each word as a mnemonic device.**

__co__mutative: use the __co__ for __c__hang up __O__rder

__So__ciative: use the __asso__ for __A__always __S__be there __S__love __O__rder

In conclusion: Remembering that these are “Manipulation” Properties that allow us __exchange__ the order of operations, then the commutative property __changes__ the *order* of numbers and the associative property __changes__ the *grouping* of the numbers

Also note that (1) these properties can only be used if all operation symbols (+ ox ) are equal, (2) if operations are mixed, fall back to PEMDAS, and (3) these properties are TRUE for addition ONLY and multiplication. **NEVER** do any of these properties with subtraction or division. **NEVER**.

We have one more “Manipulation” property to cover, but before we do, I want you to read it again until you can explain it to someone else without looking at it; And, you can explain to someone else how commutative and associative properties are SIMILAR and how they are DIFFERENT.

The reason I keep encouraging speaking out loud is because humans are very good at joking inside our heads that we understand something, but it’s almost impossible to say it out loud if we don’t really know it. Talking out loud keeps us honest with ourselves. The reason I encourage telling someone else is (1) if you really don’t know, you can’t tell someone else, and (2) teaching someone else is the fastest way to learn things yourself Now, go practice.

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