BillGatesJR
2008-07-03, 22:28
I was hoping to submit a math help article I wrote to help the community. Please review what I wrote below and tell me if it will be helpful or if it is just a waste of time to post. I know a lot of people who have trouble in mathematics (namely algebra), so do you think this will be of any help? I checked it, there should be no errors but if you spot any I'd greatly appreciate your help. Here is the article I hope to submit:
THE ORDER OF OPERATIONS EXPLAINED
I'm sure you have heard of "Please Excuse My Dear Aunt Sally" or "PEMDAS" in your high school math class, as a way of helping you remember what is known as the Order of Operations.
Basically the Order of Operations is self-explanatory -- it is a set of rules that determine the order in which mathematical functions should be used in a complex algebraic equation. This article will explain why the rule is necessary, so that it will be easier to understand and use the next time you are stuck on a tricky math problem.
Lots of people have trouble with mathematics (especially in this area) so I decided I should help out the community by breaking this down into baby steps for the reader. Using simplistic examples, you will see exactly why this rule comes into play.
-----------------
The Order of Operations is as follows:
Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
[PEMDAS or "Please Excuse My Dear Aunt Sally" should help you to memorize it.]
MULTIPLICATION/DIVISION AND ADDITION/SUBTRACTION
Let's start with these two groups, since they are the easiest to explain. This section will show why multiplication/division must come before add/subtraction. One of my favorite (and easiest) ways of modeling examples for mathematical situations is using apples and baskets.
Suppose you have 2 baskets. Each basket is large enough to carry 10 apples. The first basket is full, but the other basket is missing one apple. How many apples do you have? First, let me point out the blatantly obvious: this problem can be solved using one simple operation, addition.
10 + 9 = 19 apples
Easy right? However, in order for me to prove why the PEMDAS rule is necessary, I will have to complicate things a bit by solving this problem using 2 mathematical steps instead of just one. I shall construct an equation that will result in the same solution, which is 19 apples.
c*b-m
c = carrying capacity of each basket (10 apples)
b = number of baskets (2 baskets)
m = missing apples (1 missing)
Now we just have to plug in the numbers from the story problem into the equation:
10*2-1
Step 1: 20-1
Step 2: 19
Result: 19 apples
Now do you see why multiplication must come first? If not, let's point out why. I'll do the operations in reverse:
10*2-1
Step 1: 10*1
Step 2: 10
Result: 10 apples
Wrong! If we counted the apples one by one, we would see that there are 19 apples, therefore this way of solving the problem is not mathematically sound. This proves that MDAS is the correct order. But what about parantheses and exponents? Those will be covered next. Please note that multiplication is not done BEFORE division, nor addition BEFORE subtraction, the operations are grouped together. You perform them left to right, doing mult./div., then doing add./subt. Here is an example:
4-2+3/1*5
Step 1: 4-2+3*5 then 4-2+15
Step 2: 2+15 then 17
Result: 17
PARENTHESES AND EXPONENTS
Now let's show why parentheses have to be taken care of before exponents. Back to the original apple basket situation:
10 apples per basket * 2 baskets - 1 missing apple
Again, let us further complicate this by including parentheses and exponents into the mix.
(5+2)*2^1-1
Step 1: (10)*2^1-1
Step 2: 10*2-1
Step 3: 20-1
Step 4: 19
Result: 19 apples
If you get any number other than 19, the operations were not performed in the correct order. This proves that PEMDAS is correct. And...there you go! Good luck in math class!
THE ORDER OF OPERATIONS EXPLAINED
I'm sure you have heard of "Please Excuse My Dear Aunt Sally" or "PEMDAS" in your high school math class, as a way of helping you remember what is known as the Order of Operations.
Basically the Order of Operations is self-explanatory -- it is a set of rules that determine the order in which mathematical functions should be used in a complex algebraic equation. This article will explain why the rule is necessary, so that it will be easier to understand and use the next time you are stuck on a tricky math problem.
Lots of people have trouble with mathematics (especially in this area) so I decided I should help out the community by breaking this down into baby steps for the reader. Using simplistic examples, you will see exactly why this rule comes into play.
-----------------
The Order of Operations is as follows:
Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
[PEMDAS or "Please Excuse My Dear Aunt Sally" should help you to memorize it.]
MULTIPLICATION/DIVISION AND ADDITION/SUBTRACTION
Let's start with these two groups, since they are the easiest to explain. This section will show why multiplication/division must come before add/subtraction. One of my favorite (and easiest) ways of modeling examples for mathematical situations is using apples and baskets.
Suppose you have 2 baskets. Each basket is large enough to carry 10 apples. The first basket is full, but the other basket is missing one apple. How many apples do you have? First, let me point out the blatantly obvious: this problem can be solved using one simple operation, addition.
10 + 9 = 19 apples
Easy right? However, in order for me to prove why the PEMDAS rule is necessary, I will have to complicate things a bit by solving this problem using 2 mathematical steps instead of just one. I shall construct an equation that will result in the same solution, which is 19 apples.
c*b-m
c = carrying capacity of each basket (10 apples)
b = number of baskets (2 baskets)
m = missing apples (1 missing)
Now we just have to plug in the numbers from the story problem into the equation:
10*2-1
Step 1: 20-1
Step 2: 19
Result: 19 apples
Now do you see why multiplication must come first? If not, let's point out why. I'll do the operations in reverse:
10*2-1
Step 1: 10*1
Step 2: 10
Result: 10 apples
Wrong! If we counted the apples one by one, we would see that there are 19 apples, therefore this way of solving the problem is not mathematically sound. This proves that MDAS is the correct order. But what about parantheses and exponents? Those will be covered next. Please note that multiplication is not done BEFORE division, nor addition BEFORE subtraction, the operations are grouped together. You perform them left to right, doing mult./div., then doing add./subt. Here is an example:
4-2+3/1*5
Step 1: 4-2+3*5 then 4-2+15
Step 2: 2+15 then 17
Result: 17
PARENTHESES AND EXPONENTS
Now let's show why parentheses have to be taken care of before exponents. Back to the original apple basket situation:
10 apples per basket * 2 baskets - 1 missing apple
Again, let us further complicate this by including parentheses and exponents into the mix.
(5+2)*2^1-1
Step 1: (10)*2^1-1
Step 2: 10*2-1
Step 3: 20-1
Step 4: 19
Result: 19 apples
If you get any number other than 19, the operations were not performed in the correct order. This proves that PEMDAS is correct. And...there you go! Good luck in math class!