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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 (that’s my name) provide the rules for working with (manipulating) numbers and/or terms and allow us to currency the order of operations. The first property we discussed was the commutative property of addition/multiplication. It allows us to change the order 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 a (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 numbers does NOT change – we only change the grouping. We move the ( ). Why? Again, it’s about changing the Order of Operations if there’s a reason to do it. For example, in 3 + (17 + 19), the correct working order would be to do the ( ) first. But I need my calculator for (17 + 19). That should not need a calculator, but many students are addicted to the calculator and grab this calculator too quickly. If they just took a second to actually 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 . Whenever we can eliminate the use of the calculator, we save time.
As before, to use the associative property, all operations must be the same – all additions or all multiplications. And as before, if the operations are mixed, you must use PEMDAS.
Stop whining! I hear you complaining! “Should I remember names? “How can I remember which property does what? Of course I can’t really hear you moans; but that’s traditionally where algebra students really start whining with exactly those questions. And the answers are…
Yes, you also need to learn names and spell them correctly. Yes, I was doing spelling tests in algebra class. You should have heard the uproar that cause ! “It’s not an English class. ‘Why do we have to spell in math class?’ It was kind of funny, except they really believed they didn’t need to know how to spell a word no used in English classes. I think correct spelling is important in all classes. I know that being able to spell a word correctly helps with pronunciation and vice versa. Far too many students pronounce the commutative as if it was communative. There is NO ‘n’ in the commutative and the stem of the word is commute – not common. So practicing spelling and pronunciation aloud is beneficial.
The reason knowing the correct names is important is the same reason we all have names. Knowing the names is much faster than having to describe everything. It’s like asking about your child’s friend, Joey, rather than having to describe the little freckled, red-haired boy who lives three blocks away. Names are such time savers.
As for how to remember which property does what… I’ll give you two ways to remember each one. Then choose the method that works best for you.
1st method: By root word.
The root of the commutative word is commute. Like going to school or work: Colorado Springs to Denver in the morning and Denver to Colorado Springs in the evening. The order towns is different, but the distance is 60 miles each way.
The root of the associative word is associated. Who do you associate with? Who’s in your band? The ( ) in these problems change which numbers are grouped or matched first.
2nd method: Use the first letters of each word as a mnemonic device.
Commutative: use the co to VSchange Oorder
Associationcitation: use the associated to Astill Sstay inside Sme Oorder
In conclusion: By recalling that these are “Manipulation” Properties that allow us currency the order of operations, then the commutative property changes them order numbers and the associative property changes them grouping numbers.
Also remember that (1) these properties can only be used if all operation symbols ( + or x ) are the same, (2) if the operations are mixed, fall back to PEMDAS, and (3) these properties are ONLY TRUE for addition and multiplication. NEVER do either of these properties with subtraction or division. NEVER.
We have one more property of “manipulation” to cover, but before that, I want you to re-read this until you can explain it to someone else without looking; 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 that we humans are very good at making ourselves think we understand something, but it’s nearly impossible to say it out loud if we don’t. not really. Speaking out loud keeps us honest with ourselves. The reason I encourage telling someone else is (1) if you don’t really know, you can’t tell someone else, and (2) teach someone another is the fastest way to learn things yourself. Now go practice.
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