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 to change 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 when 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 (b x c) = (a x b) x c (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 than the commutative property? With the associative property, the order of the numbers does NOT change--we only change the grouping. We move the ( ). Why? Again, it is to change the Order of Operations if there is a reason to do so. For example, in 3 + (17 + 19), the correct working order would be to do the ( ) first. But I need my calculator for (17 + 19). This shouldn't need a calculator, but many students are calculator dependent and grab that calculator too quickly. If they just take a second to actually look, they would see that applying the associative property changes the problem to (3 + 17) + 19. No calculator needed here because 3 + 17 = 20 and 20 + 19 is 39. Likewise, 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. Any time that we can eliminate calculator usage, we have saved time. As before, in order to use the associative property, all of the operations must be the same--all addition or all multiplication. And as before, if the operations are mixed, you must use PEMDAS. Stop that whining! I can hear you complaining! "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 actually DO start whining with exactly those questions. And the answers are... Yes, you have to learn the names and SPELL them correctly as well. Yes, I used to give spelling tests in Algebra class. You should have heard the uproar that caused! "This isn't English class. " Why do we have to spell in Math class?" It got to be kind of funny, except for the fact that they truly believed that they didn't need to know how to spell any word not used in English class. 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 commutative as if it were communative. There is NO 'n' in commutative and the root word is commute--not commune. So spelling and out loud pronunciation practice are both beneficial. The reason that knowing the correct names is important is the same reason we all have names. It is much quicker to know the names than to have to describe everything. It's like asking about your child's friend, Joey, rather than having to describe the short little boy with freckles and red hair who lives three streets over. Names are such time savers. As to how to remember which property does what...I am going to give you two ways to remember each one. Then choose the method that makes the most sense to you. 1st method: By root word. The root word of commutative is 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 of associative is associate. With whom do you associate? Who is in your group? The ( ) in these problems change which numbers get grouped or associated first. 2nd method: Use the first letters of each word as a mnemonic device. Commutative: use the co for Change Order Associative: use the asso for Always Stay in Same Order In conclusion: Remembering that these are "Manipulation" Properties which let us change the Order of Operations, then the commutative property changes the order of the numbers and the associative property changes the grouping of the numbers. Also remember that (1) these properties can only be used if all of the operation symbols ( + or x ) are the same, (2) if operations are mixed, fall back on 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 "Manipulation" Property to cover, but before we do 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 the commutative and associative properties are SIMILAR and how they are DIFFERENT. The reason the I keep encouraging speaking out loud is because we humans are very good at kidding ourselves within our heads that we understand something, but it is nearly impossible to say it out loud if we don't really know it. Speaking out loud keeps us honest with ourselves. The reason I encourage telling someone else is (1) if you don't really know it, you can't tell someone else, and (2) teaching someone else is the quickest way to learn things yourself. Now, go practice.