Michael Salas

Order of Operations

1. Purple Math- the order of operations

- Website- http://www.purplemath.com/modules/orderops.htm

- Hosted by Elizabeth Stapel

- This website is a very good resource for students and even

teachers. This website directly explains the steps of order of

operations. It clarifies on what not to do in the process of order of

operations. This website gives you the techniques on how to

approach a problem regarding the order of operations.

2. Math- The Order of Operations

- Website-

http://www.math.com/school/subject2/lessons/S2U1L2GL.html

- Hosted by math.com, (The World of Math online)

- This website is a very useful resource for students. This website

first explains the basics of order of operations. Next, the website

goes more in depth on the rules of PEMDAS or order of

operations. The next step of this website is that it shows examples

of how solve problems regarding the order of operations. Finally,

you can work out the problems that are given to you and then find

out if you are correct or incorrect.

3. Math Goodies- Order of Operations

- Website-

http://www.mathgoodies.com/lessons/vol7/order_operations.html

- Hosted by Mrs. Glosser’s Math Goodies

- This website is another useful tool for students. This website

carefully explains the wrong and right way to do a problem. By

telling you the wrong way to do it, it helps the student understand

that is not the way to do it. It explains how PEMDAS operates.

This website goes step by step on how to approach a problem.

4. Classroom.jc-schools- Order of Operations

- Software- Math Order of Operation

http://classroom.jc-schools.net/basic/math-order.html

- Hosted is not clarified

Michael Salas

- This software is a very useful reference for students. This software

allows students to play games involving the order of operations.

You can play a basketball game or building a snowman game

while learning the order of operations. I believe students would

love to play this software.

5. A to Z Teacher Stuff- Order of Operations

- Internet Project- An Educator’s Desk Lesson Plan-

http://atozteacherstuff.com/go/search.cgi?query=order+of+operatio

ns&grade=&bool=OR

- Written by Bambi Exum

- This is a great reference for students and teachers. There are many

links to help you teach the order of operations. You are able to

create your own lesson plan and submit it on this internet project.

Michael Salas

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The Order of Operations: PEMDAS (page 1 of 3)

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!":

Choice 1: 4 + 2×3 = (4 + 2)×3 = 6×3 = 18

Choice 2: 4 + 2×3 = 4 + (2×3) = 4 + 6 = 10

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, established at least as far back as the 1500s, called the "order of operations". The "operations" are addition, subtraction, multiplication, division, exponentiation, and grouping; the "order" of these operations states which operations take precedence (are taken care of) before which other operations.

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". 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 the "4 + 2×3" question at the beginning of this article, Choice 2 was the correct answer,

because we have to do the multiplication before the addition.

(Note: Speakers of British English often instead use "BODMAS", which stands for "Brackets, Orders, Division and Multiplication, and Addition and Subtraction". Since "brackets" are the same as parentheses and "orders" are the same as exponents, the two acronyms mean the same thing.)

The order of operations was settled on in order to prevent miscommunication, but PEMDAS can generate its own confusion; some students sometimes tend to apply the hierarchy as though all the operations in a problem are on the same "level", but often those operations are not "equal". Many times it helps to work problems from the inside out, rather than left-to-right, because often some parts of the problem are "deeper down" than other parts. The best way to explain this is to do some examples:

Michael Salas

Simplify 4 + 32.

I need to simplify the term with the exponent before trying to add in the 4:

4 + 32 = 4 + 9 = 13

Simplify 4 + (2 + 1)2.

I have to simplify inside the parentheses before I can take the exponent through. Only then can I do the addition of the 4.

4 + (2 + 1)2 = 4 + (3)

2 = 4 + 9 = 13

Simplify 4 + [–1(–2 – 1)]2.

I shouldn't try to do these nested parentheses from left to right; that method is simply too error-prone. Instead, I'll try to work from the inside out. First I'll simplify inside the curvy parentheses, then simplify inside the square brackets, and only then take

care of the squaring. After that is done, then I can finally add in the 4:

4 + [–1(–2 – 1)]2

= 4 + [–1(–3)]2

= 4 + [3]2

= 4 + 9 = 13

There is no particular significance in the use of square brackets (the "[" and "]" above) instead of parentheses. Brackets and curly-braces (the "{" and "}" characters) are used when there are nested parentheses, as an aid to keeping track of which parentheses go with which. The different grouping characters are used for convenience only. This is similar to what happens in an Excel spreadsheet when you enter a formula using parentheses: each set of parentheses is color-coded, so you can tell the pairs: Copyright © Elizabeth Stapel

2000-2011 All Rights Reserved

Simplify 4( –2/3 + 4/3 ).

I will simplify inside the parentheses first:

Michael Salas

Then I get 4(–2

/3 + 4/3) =

8/3

The next page has more worked examples examples....

Home | Teacher | Parents | Glossary | About Us

Michael Salas

When expressions have more than one operation, we have to follow rules for the order of operations:

1. First do all operations that lie inside parentheses. 2. Next, do any work with exponents or radicals. 3. Working from left to right, do all multiplication and division. 4. Finally, working from left to right, do all addition and subtraction.

Click Show Me to see the correct order to solve the expression below:

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Order of

Operations

Unit

7 >

Less

on 1

of 8

Problem

:

Evaluate the following

arithmetic expression:

3 + 4 x 2

Solution

: Studen

t 1

Studen

t 2

3 + 4 x

2

3 + 4 x

2

= 7 x 2 = 3 + 8

= 14 = 11

Michael Salas

It seems that each student interpreted the problem differently,

resulting in two different answers. Student 1 performed the

operation of addition first, then multiplication; whereas student

2 performed multiplication first, then addition. When performing

arithmetic operations there can be only one correct answer.

We need a set of rules in order to avoid this kind of confusion.

Mathematicians have devised a standard order of operations

for calculations involving more than one arithmetic operation.

Rule 1: First perform any calculations inside parentheses.

Rule 2: Next perform all multiplications and divisions, working

from left to right.

Rule 3: Lastly, perform all additions and subtractions, working

from left to right.

The above problem was solved correctly by Student 2

since she followed Rules 2 and 3. Let's look at some

examples of solving arithmetic expressions using these

rules.

Example 1: Evaluate each expression using the rules for order of

operations.

Solution: Order of Operations

Expression Evaluation Operation

6 + 7 x 8 = 6 + 7 x 8 Multiplication

= 6 + 56 Addition

= 62

16 ÷ 8 - 2 = 16 ÷ 8 - 2 Division

Michael Salas

= 2 - 2 Subtraction

= 0

(25 - 11) x 3 = (25 - 11) x 3 Parentheses

= 14 x 3 Multiplication

= 42

In Example 1, each problem involved only 2 operations. Let's

look at some examples that involve more than two operations.

Example

2:

Evaluate 3 + 6 x (5 + 4) ÷ 3 - 7 using the order of

operations.

Solution: Step

1:

3 + 6 x (5 + 4) ÷ 3

- 7

=

3 + 6 x 9 ÷ 3

- 7

Parenthes

es

Step

2:

3 + 6 x 9 ÷ 3 - 7 =

3 + 54 ÷ 3 - 7 Multiplicati

on

Step

3:

3 + 54 ÷ 3 - 7 =

3 + 18 - 7 Division

Step

4:

3 + 18 - 7 =

21 - 7 Addition

Step

5:

21 - 7 =

14 Subtractio

n

Michael Salas

Michael Salas

Operation Order 4 Operations Quiz Mystery Picture

Order of Operation

Order of Operations

Basketball

Paint Brush Math

Operation Snowman Great Equations Math Frog

Order of Operation

Calculation

Order of Operations

Order of Operation

Flash Cards

Software evaluation form

General

Information

Title of

program

Math Order

of Operation

Subject

Areas

Math

Grade 6th

-8th

Advantages Works

properly

Useful in

lessons

Easy to

understand

Games basketball snowman Math Frog

Tutorial Yes

Drill and

Practice

Yes

Strongly Agree Agree Disagree

The directions are clear and easy to follow for the intended age group. Strongly Agree

Useful directions are available via a written manual or on-screen help. Agree

It is easy for the students to exit the program. Strongly Agree

Michael Salas

Students may start up where they left off in the program. Disagree

Searching for certain items is simple and complex searches may be done. Agree

The program functions well on the computer and it is free on bugs. Strongly Agree

The program is enjoyable to use. Strongly Agree

Included graphics and animations are meaningful. Agree

Sound can be turned off/on. Strongly Agree

The program automatically branches based on student responses. Disagree

The user level can be set by the teacher. Agree

The software contains a useful management tool for the teacher. Disagree

The software covers a wide range of skill levels. Agree

Other thoughts: These games make math fun to do instead of hearing a teacher lecturing the

whole class. These games are enjoyable to play and I hope to use these games for my future

students.

Michael Salas

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Lesson Plans : Mathematics : Grades 6-8 : Operations

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Order of Operations

Lesson Plan

A lesson teaching the order of operations that must be followed

when solving equations. Exercise and answer key attached. http://www.lessontutor.com/eesA4.html

Michael Salas

FIND MORE

Grade Level(s): 9-12 • Rating: 6 Votes: 10 Rate It

Dr. Seuss Activities

Lesson Plan

Cross-curricular activities for 15 Dr. Seuss books, including The Foot

Book, Dr. Seuss' ABC, If I Ran the Zoo, The Lorax, and more! http://www.atozteacherstuff.com/pages/250.shtml

Grade Level(s): K, 1-2 Rate It

Lunch Count and Attendance All-In-One

Teacher Tip

Since I use a racing theme in my class room, I have a race car with

the kid's names on them. Each car has a small piece of magnet on

the back and they hang on the file cabinet. Each day, the kids come in

the room and they move their race car underneath their lunch choices

(choice 1, choice 2 or lunch from home). It's a three-fold tool-I know

who is at school, I know what they want for lunch, and it's a great way

for them to show responsibility. After the first month or so, if a student

is present and they don't move their race car, I order their lunch for

them. I usually only have to do that once because they like to order

their own lunch. This would work for any theme in a classroom... just

change the race car to whatever cut-out you want. http://www.atozteacherstuff.com/pages/819.shtml

Grade Level(s): K, 1-2, 3-5 Rate It

How to make sure everyone has turned in their

paper

Teacher Tip

During the first week of school I assign everyone a number. I usually

just go down the roll book and assign the numbers in alphabetical

order. I instruct the students that each time they turn in a paper the

number should be in the right hand corner of the paper. When all

papers are collected I put the in order and I can quickly check and

see if anyone has not turned a paper in. This works great and is fast. http://www.atozteacherstuff.com/pages/1744.shtml

Grade Level(s): Rate It

April Fool's Day Printables

Printable

Worksheets for April Fool's: ABC order, vocabulary word jumble,

word searches http://www.apples4theteacher.com/holidays/april-fools-day/printables/

Michael Salas

Grade Level(s): 1-2, 3-5 Rate It

Earth Day Printables and Worksheets

Printable

Word search, ABC order, word jumble, and acrostic poem for Earth

Day. http://www.apples4theteacher.com/holidays/earth-day/printables/

Grade Level(s): K, 1-2, 3-5 Rate It

Color Patterns

Lesson Plan

To be able to make a pattern of three colors and then repeat the

pattern on paper. http://www.atozteacherstuff.com/pages/365.shtml

Grade Level(s): Preschool, K Rate It

Numbers

Teacher Tip

To emliminate always having to write new studnets names on small

pieces of paper or whatever I was using at the time, I simply

numbered my students and gave them each a clothes pin. Then I

chose 5 needed jobs. I wrote the titles on sentence strips and placed

them in a small apple pocket chart (any pocket chart will do- small is

the key). I clipped the clothes pin next to the title and then I move

them each week in number order. All students not working that week

are 'on vacation.' Any unused clothes pins go on top at the leaf. No

more need to remember who did what last week or what about the

new student. http://www.atozteacherstuff.com/pages/1648.shtml

Grade Level(s): Rate It

Sentence Strip Unscramble

Teacher Tip

This activity is great to use the first day kids walk in, while you greet

new students and their parents. Type a short letter attached to an

envelope inviting students to unscramble the sentence you have

placed in their envelope. Sentences like: Welcome to 3rd grade! It's

going to be a great year! I'm glad you are here! Be sure to cut

between the words like a puzzle. The students job is to place the

sentence in order. http://www.atozteacherstuff.com/pages/5069.shtml

Grade Level(s): 1-2, 3-5 Rate It

Michael Salas

Counting by 2: Fill in the Blank (Random)

Printable

Counting by 2s - Fill in the blank 2 to 100 (5 problems in random

order) math/countingby2s_2.pdf

Grade Level(s): 1-2 Rate It

Everyone gets a turn

Teacher Tip

In order to make sure all students are given an opportunity to answer,

read or respond in my classroom, I purchased an inexpensive deck of

playing cards and wrote a student's name on each card, making sure

to use each name an equal number of times. Before each class, I

shuffle the cards. I hold them, name up, and use them to call on

students. It adds ramdomness and keeps students alert because they

do not know when they will be called on. http://www.atozteacherstuff.com/pages/4481.shtml

Grade Level(s): 5-7 Special Education Rate It

Storing Sentence Strips

Teacher Tip

I use a lot of sentence strips in my classroom and have found that

shower curtain hooks used in a hole punched hole keep all for a

particular subject in order. These can be stored in a wallpaper trough,

or I hang them on a line. http://www.atozteacherstuff.com/pages/1771.shtml

Grade Level(s): Rate It

Subject/Verb Agreement and Pronoun/Antecedent

Agreement

Lesson Plan

This lesson is designed to offer a hands on approach to learning

pronoun/antecedent agreement--analyzing sentences and identifying

pronouns and their antecedents in order to verify their agreement. http://www.atozteacherstuff.com/pages/336.shtml

Grade Level(s): 6-8 • Rating: 4 Votes: 6 Rate It

Campaign 2004: Classroom Electorate

Lesson Plan

Students take on the role of a political analyst, forecasting the

electoral vote count for the 2004 presidential election. In order to

Michael Salas

make a prediction, students are introduced to campaign issues, the

Electoral College, the role of swing states, and the importance of

political participation. The lesson plan includes a fun, interactive

classroom competition where students make electoral predictions and

compare with the actual results following the November 2nd voting

deadline. http://www.atozteacherstuff.com/pages/4533.shtml

Grade Level(s): 6-8, 9-12 Rate It

Working Together for Classroom Behavior...

Teacher Tip

In order to motivate the entire class to work together for great

classroom behavior, I am designing a race track with several pit rows

on poster board. Whenever the class gets a compliment from the

principal, a parent, an assistant, or teacher, the race car will be

advanced one space. If group misbehavior continues aft... more>> http://www.atozteacherstuff.com/pages/1733.shtml

Grade Level(s): • Rating: 8 Votes: 6 Rate It

Counting by 3: Fill in the Blank (Hints)

Printable

Counting by 3s (3 to 120 in order w/ HINTS) Fill in the blank math/countingby3s_2.pdf

Grade Level(s): 1-2 Rate It

TeachDelaware

Resource Site

Delaware Center for Teaching Careers (TeachDelaware) was

established by Delaware State Department of Education to recruit

qualified individuals to the teaching profession in order to alleviate

the shortage of credential teachers in Delaware. TeachDelaware is a

one-stop information and referral recruitment center for individuals

who may be interested in a teaching career. TeachDelaware offers a

variety of informational and advisor-assisted services to prospective

teachers. http://www.teachdelaware.com/

Grade Level(s): Rate It

FOCUS on Classroom Management: Week 1 -

Focus Attention on You!

Resource Site

All too often, the most common mistake many teachers make is

Michael Salas

starting a lesson without all of the students' attention. Eager to get

into teaching, a teacher might begin the lesson while some students

are visiting with their neighbor, searching through their desks, or even

walking around the classroom! http://www.atozteacherstuff.com/pages/7664.shtml

Grade Level(s): K, 1-2, 3-5, 6-8, 9-12 Rate It

Counting by 3: Fill in the Blank (Random w/ Hints)

Printable

Counting by 3s (3 to 102, 5 problems in random order w/ hints) Fill in

the blank math/countingby3s_4.pdf

Grade Level(s): 1-2 Rate It

Lesson Plan

The purpose of this lesson is to create an awareness and

understanding that important technological advances within our

culture, in a variety of areas, are the result of the work done by a

diverse group of people, many of whom are African Americans. To

this end, students will be actively involved in discussions, research,

and sharing sessions. Between six and ten class periods will be

necessary for this lesson. http://www.atozteacherstuff.com/pages/252.shtml

Grade Level(s): 6-8 • Rating: 8 Votes: 9 Rate It

Mexican Food

Lesson Plan

Following a lecture on the history of Mexican food, grade five students

will be able to recognize and order Mexican cuisine, at a restaurant,

with 100% accuracy. http://www.atozteacherstuff.com/pages/368.shtml

Grade Level(s): 3-5 Rate It

Using student numbers

Teacher Tip

Because keeping up with papers from each student in each subject

can be overwhelming, I've added onto the idea of assigning student

numbers in the classroom. When students head their papers, they are

required to include their personal numbers based on alphabetical

order. At the end of the day, papers are collected and put in "ordinal

position" by students. I have a quick record of which papers are

missing. When papers are graded, student helpers file the papers in

Michael Salas

individual file folders for the students to check. The folders are kept in

a basket in the back of the room and I don't waste valuable time

passing papers to students. http://www.atozteacherstuff.com/pages/1750.shtml

Grade Level(s): Rate It

Filing System

Teacher Tip

I have found a wonderful way to file all my papers. It really cuts down

on time and desk clutter and anybody can do it. Assign each subject a

color. For example, red for reading, yellow for math, green for

science, etc. Buy (I hate that word too, but they are not expensive), or

get a parent to donate, file folders of each color for each subject. I

started out with at least 5 of each color. Then file and label the

various topics for each subject in the corresponding color. This way if

you need a worksheet on clocks, you know that all the math files are

yellow and you go to the yellow folder labled time. If you need a

reading comprehension test sheet, you would look in the red folder

labled comprehension tests. It is also an ideal system to use for

memos from the office and order forms from various companies. I

hope it works for you as well as it has for me! http://www.atozteacherstuff.com/pages/1784.shtml

Grade Level(s): Rate It

Family Tree

Lesson Plan

Students will examine their family history in order to create a family

tree. http://www.canteach.ca/elementary/wsoccult4.html

Grade Level(s): 3-5 • Rating: 5 Votes: 8 Rate It

Mixed Up Sentences

Lesson Plan

I mix up words using those sight words being worked on and words

from our word walls. The children then put the words in order to make

a sentence. http://www.atozteacherstuff.com/pages/414.shtml

Grade Level(s): K, 1-2 Rate It

Pages: 1 2 [>>]

Michael Salas

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