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A Parents Guide To Algebra’s Basic Concepts – Properties Made Easy – Commutative Property
With this article, we enter the first of the main problem areas of algebra – properties. These properties are fundamental to both algebra and geometry, but many students start having trouble at this point because they have trouble telling the properties apart and because the “why do I have to -I need to know this” begins.
Students’ difficulty with algebra is easier to understand when we realize that until algebra, students are working on math skills that they see being used regularly around them. Students may have struggled to memorize multiplication facts or work with fractions, but they understood that these skills were necessary. With algebra, students are introduced to topics they have never seen before at a much faster pace than they have encountered before.
Unfortunately, another cause of this difficulty is again the problem of “insufficient explanation”. Just as many instructors start teaching algebra topics without ever explaining what algebra is; too many teachers persist in presenting the properties without sufficient explanation of Why They are important. This happens for two main reasons: (1) the push to cover skills quickly caused by No Child Left Behind, and (2) the assumption that presenting the material once means students understand. They don’t.
I know that even the mention of the word PROPERTIES makes some people want to yawn and others want to run for the hills. But please don’t either. We are very, very close to starting with the most basic concepts of algebra – solving equations. These properties give us the rules and the basis for successfully solving equations. (Wow! The word “successfully” has Three sets of double letters. Costs!)
In order to make the properties easier to understand and remember, I will divide them into three different groups based on their uses. The first group of properties I will call the “manipulation” properties. You will never see this name in any algebra book – I made it up because I think it helps explain the purpose of this group of properties. This is how Shirley Slick remembers these 3 properties: the commutative property, the associative property and the distributive property. These properties tell us what we are allowed to do to “manipulate” or work with numbers or and/or terms. Moreover, they allow us to change the order of operations. (Remember PEMDAS?)
The first of the “Manipulation” properties: Commutative property for addition/multiplication
In symbols, this property says: a + b = b + a (addition) or axb = bxa (multiplication) (The x here means multiplication — not a variable. It can also be written ab = ba or (a)( b ) = (b)(a) All mean multiplication.) Note: This property is NOT true for subtraction or division.
With numbers, the commutative property looks like: 4 + 7 = 7 + 4 or (5)(6) = (6)(5)
So what’s the point? Remember that the order of operations says that we to have to add from left to right and we to have to multiply from left to right; but the commutative property says not necessarily. He says that if there’s a reason to do ityou can change order around. For example: the problem 6 + 39 + 4 is an easier problem if we change order to 6 + 4 + 39. It’s easier because 6 + 4 is 10 and 10 + 39 = 49. For multiplication, and the example might look like 25 x 7 x 4. UGH! Where’s my calculator? But wait… If we change the orderr to 25 x 4 x 7 and remember that 4 quarters is a dollar, then 25 x 4 = 100 and 100 x 7 = 700. This method is very simple and faster than getting out a calculator.
There are 2 important conditions to remember: (1) in order to use this property, all “operation symbols” (+ or x) to have to be them same–either all additions or all multiplications, or (2) if you have mixed operations, like 25 x 6 + 4, then you to have to follow the normal order of operations (PEMDAS).
In conclusion, the commutative property for addition and the commutative property for multiplication allow us to change order when adding or multiplying. Now read this property over and over – out loud – until it makes sense to you and you can explain it to someone else without looking back.
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