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Order of Operations Worksheets

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  1. Algebra Word Problems
  2. Algebraic Solutions to Simultaneous Equations
  3. Associative Property
  4. Combining Like Terms (Difficult)
  5. Combining Like Terms (Simple)
  6. Commutative Property
  7. Distributive Property
  8. Evaluating Formulas
  9. Like Terms
  10. Open Ended Integer Problems
  11. Order of Operations (2-step problems)
  12. Order of Operations (3-step problems)
  13. Order of Operations (4-step problems)

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PEMDAS - Order of Operations

What does order of operations mean?

Order of operations is a ranking of different mathematical procedures, first to last. You can think of this as a sort of recipe on how to do things. An order of operations for getting dressed might have the rule: Socks before shoes, socks and shoes before pants.

The letters, PEMDAS, stand for Parenthesis, Exponents, Multiplication, Division, Addition and Subtraction. Parenthesis is the operation you do first, and Subtraction the one you do last.

To show you how important order is, look at the following sentences, where the words are all the same, but the order is different:

  • The bear ate Mary and Jane's lunch on the table.
  • The bear ate Mary, and Jane's lunch on the table.
  • The bear on the table ate Mary and Jane's lunch.
  • Mary and Jane's bear ate the lunch on the table.

It gets pretty hard to tell who is eating what.

This is the same reason order of operations is so important in mathematics. Mathematics relies on clear communication and without this rule the meaning of statements would get confused.

How is PEMDAS used?

One common application is in computer science. When a computer program is written, PEMDAS order is used by the computer. While the computer will follow the proper order, sometimes programmers will make mistakes.

Here is a line of computer code: share = tips/workers+1

It is part of a program used to manage money at a coffee shop. What the programmer intended was, "Each worker's share of the tips is equal to the total amount of tips divided by the number of servers working that shift (workers +1 where the plus one is for the guy at the drive-through window).

His program will fail because the computer will do the division, tips/workers first (using the rule that division comes before addition) and then add 1.

It might be even worse than just giving the wrong answer. If the counter is closed (and only the drive-through is open) then workers will be zero and the computer will try to divide tips by zero, an illegal operation that might crash the computer.

In this example, the correct code would be: share = tips/(workers + 1) where the parenthesis now make everything work properly.

A basic problem with order of operations.

What is 15 - 5 + 4 ?

Here, the order of operations, PEMDAS, seems to tell us to do addition before subtraction. Which would give us 5 + 4 first, then 15 - 9 next, and our answer would be 6. But addition and subtraction (as well as multiplication and division) actually have the same precedence, or order.

This can be confusing, but operations with the same order are done left to right, and the correct method would be: first do 15 - 5 to get 10, then add 4 to get 14.

An interesting fact about order of operations.

It may surprise you to learn that our bodies have an internal 'order of operations' built into them. What comes first affects what comes next.

If you hold a heavy weight for a few minutes, and then pick up something much lighter with the same hand, your perception will be that the second object is even lighter than it actually is. This is a different experience than picking up the lighter weight first.

The same thing happens with our eyes. After our eyes have adapted to a darker environment, going into a sunlit area will make normal daylight seem much, much brighter than otherwise. This is why, when driving through a long tunnel, signs will direct drivers to turn on their headlights - even if the tunnel already has lights. They are trying to prevent a sudden, temporary shock when someone drives out of the tunnel into daylight again.


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