Lesson 1: Order of Operations

PEMDAS

If you are asked to simplify something like "4 + 2×3", the question that naturally arises is "Which way do I do this? Because there are two options!":

It seems as though the answer depends on which way you look at the problem. But we can't have this kind of flexibility in mathematics; math won't work if you can't be sure of the answer, or if the exact same problem can calculate to two or more different answers. To eliminate this confusion, we have some rules of precedence called "order of operations", the "operations" being addition, subtraction, multiplication, division, exponentiation, and grouping.

A common technique for remembering the order of operations is the abbreviation "PEMDAS", which is turned into the phrase "Please Excuse My Dear Aunt Sally" (or, as my husband prefers, "Purple Elephants May Destroy A School"). It stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". This tells you the ranks of the operations: Parentheses outrank exponents, which outrank multiplication and division (but multiplication and division are at the same rank), and these two outrank addition and subtraction (which are together on the bottom rank). When you have a bunch of operations of the same rank, you just operate from left to right. For instance, 15 ÷ 3 × 4 is not 15 ÷ 12, but is rather 5 × 4, because, going from left to right, you get to the division first. If you're not sure of this, test it in your calculator, which has been programmed with the Order of Operations hierarchy. For instance, typesetting this into a graphing calculator, you will get:

Using the above hierarchy, we see that, in "4 + 2×3", Choice 2 is correct, because we have to do the multiplication before the addition.

But PEMDAS can generate its own confusion, because students tend to apply the hierarchy as though all the operations in a problem are on the same "level", but often they're not. Many times it helps to work problems from the inside out, because often some parts of the problem are "deeper down" than other parts. The best way to explain this is to do some examples:

Square brackets "[ ]" are sometimes used when more than one set of parentheses is needed in a problem. The different grouping symbols help to distinguish which parentheses should be paired. Parentheses and square brackets have the same meaning when you are simplifying expressions using order of operations.

Here are some sample problems for you to try.

  • Simplify 14x + 5[6 – (2x + 3)].
  • Simplify the following:

      • –9 + (9 + 20)/3(5) – (–3)

  • Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.


Evaluation

"Evaluation" mostly means "plugging numbers in for variables, and simplifying". Sometimes they just give you the numbers, and want you to simplify, which is more of an order-of-operations kind of question. In this lesson, I'll concentrate on the "plug and chug" aspect of evaluation: plugging in values for variables, and "chugging" my way to the simplified answer. Usually the only hard part in evaluation is in keeping track of the minus signs. I would strongly recommend that you use parentheses liberally, especially when you're just getting started. Here are some typical examples:

Notice that this does not match the answer to the previous evaluation!

Here are some sample problems for you to try.

In these sample problems, let a = 3, b = -2, and c = -1.

  • Evaluate ac-ab.
  • Evaluate ac2 + b2c.

Click here for answers to all the sample problems.

Grading for this lesson:

  • 10 is for a perfect set of answers the first time, or after one immediate revision. (see below)
  • 9 is for 1, 2, or 3 mistakes on the original submission. In this case, you have an option: you can accept the 9, or you can post one revision within one week after the original submission was graded. If you correct all errors in this revision, your grade will be changed to a 10. In either case, you can proceed to the next lesson.

Students who miss more than 3 questions on their first posting must submit revisions until all problems are correct. Students should not submit incomplete revisions, or revisions without the work shown.

  • 9: If you missed 4 or 5 questions on your original submission, you will receive a 9 after you have answered all questions correctly. You have unlimited revision attempts, but revisions must show your work.
  • 9: If you missed more than 5 questions on your original submission, you will receive a 9 if you have answered all questions correctly within 5 revisions, not including questions that you post to the teacher.

Any questions should be substantive:
"I don't understand" doesn't help the instructor help the student; the student needs to ask specific questions. If a student cannot form a question to ask, they may instead post three examples with work shown to demonstrate what they are doing (and what they need help with) without it counting against the revisions.

  • 8 will be given if you missed more than 5 questions on your original submission and you have answered all questions correctly. You have unlimited revision attempts, but revisions must show your work.

Grades may be lowered by one point for when the student is repeatedly not paying attention to instruction, for lack of effort on the part of the student, or for rude or inappropriate responses.

  • 7 will be given when the student is repeatedly not paying attention to instruction, for continuing lack of effort on the part of the student, or for repeated rude or inappropriate responses. All revisions must still be completed.
  • 6 reserved for administrative use
  • 5 will be given for cheating and plagiarism - purposeful or mistaken which will lower your final grade for the course (so be very careful when posting your work!) Revisions and reworking may still be required of the student, even though it will not improve the grade.

No lesson is complete without the approval of the instructor, and all revisions must be completed before a grade is assigned. No grade will be given for incomplete work.

Lesson 1 Assignment

Do the test below. You must successfully complete all of the questions in order to complete the lesson.

If you get any wrong, you will be asked to resubmit the wrong answers and show your work. The teacher will then look at your work and give you advice on what you are doing wrong.

Name:
Enter your correct email address:


1. Simplify 5 + (7-1)2/6

A11
B41/6
C51/6
D7
E17/6
F23

2. Simplify (26/2 + 33)/(23)

A11/3
B5
C20/3
D8
E6
F131/8

3. Simplify 7 + 2(5 – 9)2/[(4 + 1)2 – 3(22)]

A-23/2
B18/7
C-13
D9/4
E8/41
F3

4. Simplify 36 – 2x[4(-3-2)]

A36 - 28x
B16 - 2x
C76x
D40x + 36
E-720x
F36 + 8x

5. Simplify 24 – [3(4x+2)] + 7

A-12x + 25
B31 - 14x
C13x
D37 - 12x
E93x
F4x - 3

For problems 6 – 10 let a = 2, b = -4, c = 7, and d = -3.

6. Evaluate (ac + b)/a

A3
B28
C5
D-7
E11
F7

7. Evaluate b3/a + c2

A55
B17
C-18
D81
E8
F-4

8. Evaluate bc/a2 + d

A-10
B-28
C3
D25
E4
F-7/2

9. Evaluate b – a(cd + b) + a

A506
B2
C-44
D-52
E-90
F52

10. Evaluate c + d[(ac + b)/(d-a)] + c

A-20
B36
C12
D20
E52
F-12

(HINT: For problems 11 – 15, you may find it helpful to consider common mistakes that were discussed in the lesson.)

11. Sherry simplified the expression 3 + (1 + 2)2 and obtained an answer of 8. What mistake did she most likely make?

AShe obtained an incorrect answer when she simplified the exponent.
BShe made a mistake when summing the numbers.
CShe multiplied by 3 instead of adding 3
DShe forgot to square the sum in parentheses.
EShe simplified the exponent before performing the operation in parentheses.
FShe squared all of the numbers.

12. Franz simplified the expression 17 – [2(3 + 6)/3] and obtained an answer of 5. What mistake did he most likely make?

AHe performed multiplication before taking the sum in parentheses.
BHe performed division before taking the sum in parentheses.
CHe added three and six incorrectly.
DHe cancelled the threes.
EHe subtracted from 17 incorrectly.
FHe only divided the 6 by three.

13. Cindy simplified the expression 8/2*22 – 8 and obtained an answer of -7. What mistake did she most likely make?

AShe forgot to simplify the exponent.
BShe subtracted first.
CShe subtracted incorrectly.
DShe multiplied before simplifying the exponent.
EShe multiplied before dividing.
FShe squared too many numbers.

14. Matt simplified the expression 6 – 4[5 – 3(2-1)2] + 8 and obtained an answer of 21. What mistake did he most likely make?

AHe performed the operations in order from left to right.
BHe didn’t multiply everything inside the brackets by negative four.
CHe forgot to add eight.
DHe simplified the exponent before performing the subtraction in parentheses.
EHe didn’t multiply everything inside the parentheses by negative three.
FHe dropped a negative sign.

15. Chad simplified the expression 10 – (3x + 22) and obtained and answer of -3x + 14. What mistake did he most likely make?

AHe did not simplify the exponent first.
BHe added 3 and 22.
CHe subtracted incorrectly.
DHe simplified the exponent incorrectly.
EHe forgot to square the 2.
FHe did not take the negative all the way through the parentheses.

16. The area of a rectangle may be determined by using the formula A = lw, where "A" stands for area, "l" stands for the length of the rectangle, and "w" stands for the width of the rectangle. If the length of the rectangle is 3 cm and the width of the rectangle is 5 cm, what is the area?

A2 square cm
B5/3 square cm
C8 square cm
D15 square cm
E16 square cm
F30 square cm

17. The perimeter of a rectangle may be determined by using the formula P = 2(l + w), where "P" stands for the perimeter, "l" stands for the length of the rectangle, and "w" stands for the width of the rectangle. What is the perimeter of a rectangle with the same dimensions as the rectangle in problem 16?

A15 cm
B12 cm
C11 cm
D13 cm
E16 cm
F30 cm

18. The surface area of a rectangular prism may be determined by using the formula S = 2(lw + lh + wh), where "S" stands for surface area, "l" stands for the length of the prism, w" stands for the width of the prism, and "h" stands for the height of the prism. If the length of the prism is 2 cm, the width is 4 cm, and the height is 5 cm, what is the surface area of the prism?

A76 square cm
B270 square cm
C140 square cm
D38 square cm
E200 square cm
F50 square cm

19. The area of a triangle may be determined by using the formula A = (1/2)bh, where "A" stands for area, "b" stands for the length of the base of the triangle, and "h" stands for the height of the triangle. If the base of the triangle measures 7 cm and the height of the triangle measures 4 cm, what is the area of the triangle?

A64 square cm
B14 square cm
C28 square cm
D6 square cm
E11/2 square cm
F56 square cm

20. The area of a trapezoid may be determined by using the formula A = (h/2)(a+b), where "A" stands for area, "h" stands for the height of the trapezoid, and "a" and "b" stand for the lengths of the bases of the trapezoid. If the height of the trapezoid is12 cm, a = 2 cm and b = 4 cm, what is the area of the trapezoid?

A16 square cm
B14 square cm
C24 square cm
D9 square cm
E72 square cm
F36 square cm

 



Materials on this page are copyrighted to Elizabeth Stapel
and are used with permission.