Project
Order of Operations
Objective To introduce the rules for order of operations.
www.everydaymathonline.com
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Family
Letters
Doing the Project
Recommended Use During or after Unit 7
Assessment
Management
Common
Core State
Standards
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP1, SMP5, SMP6, SMP8
Content Standards
3.OA.5, 3.OA.8
Key Concepts and Skills
• Use basic facts to solve numeric expressions.
[Operations and Computation Goals 1 and 3]
• Recognize that numeric expressions can have different values depending on the
order in which the operations are carried out.
[Patterns, Functions, and Algebra Goal 3]
Key Activities
To explore order of operations, children use two calculators, one that follows the
conventional, algebraic order of operations and one that follows a simple left-toright order of operations.
Key Vocabulary
order of operations
Materials
Student Reference Book, p. 74H
Math Masters, p. 394A
Class Data Pad
slate
Per group: 2 calculators—one that follows the
conventional order of operations (such as a
scientific calculator) and one that follows a
left-to-right order of operations (such as most
four-function calculators, including the TI-108
and Casio SL-450). See Advance Preparation.
Extending the Project
Ex
Using a calculator that follows a left-to-right order of operations to solve number
sentences, children record the key sequences that should be used to follow the
conventional, algebraic order of operations.
Materials
Children practice skills through Home Link activities.
Student Reference Book, p. 74H
Math Masters, p. 394A
Per group: 2 calculators. See Materials list in Part 1.
Additional Information
Once children begin to encounter expressions such as 5 + 3 × 2, they have to decide the order in which the indicated operations should be
carried out. The simplest order is to begin at the left and carry out the operations in the order in which they appear. This simple left-to-right
order of operations is built into many four-function calculators. But it is neither the order of operations commonly used in higher mathematics
nor the one built into more advanced calculators.
When people refer to “Order of Operations,” by convention they mean the “algebraic” order of operations in which multiplication and division
take precedence over addition and subtraction and operations of equal precedence are performed from left to right. There are other, more
specialized orders of operation that are used in technical fields but are not pertinent to school mathematics.
In Everyday Mathematics, we follow common usage and refer to the conventional order of operations as the “Order of Operations.” See Section
10.1.2 of the Teacher’s Reference Manual for further discussion.
Advance Preparation
For Part 1, gather enough calculators to provide two for each small group of 3 or 4 children. You might consider using calculators on cell
phones and on the Internet. For each group, one calculator should follow the conventional order of operations (such as a scientific calculator);
the other should follow a simple left-to-right order of operations (such as most four-function calculators including the TI-108 and Casio SL-450).
5
3
. Calculators that follow the conventional
To check whether a calculator follows the conventional order of operations, enter 6
order of operations will show 21 in the display. Calculators that follow the left-to-right order will show 33 in the display. Copy the list of rules for
order of operations from Teacher’s Lesson Guide, page 502B onto the Class Data Pad.
936
Project 7
Order of Operations
1 Doing the Project
Exploring Order of Operations
SMALL-GROUP
ACTIVITY
Tell the class that you have a mystery for them. Explain that you
wanted to solve 6 + 5 × 3. But when you used two different
calculators, you got two different answers. One calculator gave the
result as 21; the other gave 33. Write the following number
sentences on the board:
6 + 5 × 3 = 21
6 + 5 × 3 = 33
Ask children to figure out how each calculator solved the problem.
To get 21, one calculator multiplied 5 × 3 first and then added the
result to 6; 5 × 3 = 15; 6 + 15 = 21. To get 33, the other calculator
added 6 + 5 first and then multiplied the result by 3; 6 + 5 = 11;
11 × 3 = 33.
Distribute two calculators to each small group—one that follows
the conventional order of operations (such as a scientific
calculator) and one that follows a left-to-right order of operations
(such as most four-function calculators, including the TI-108 and
Casio SL-450).
Have each group use each calculator to solve 15 - 7 × 2 = ? Have
them share their answers. 16 (left-to-right) and 1 (scientific)
Ask children to explain how each of their calculators solved the
problem. Sample answers: One calculator (scientific) multiplied
first and then subtracted. The other calculator (left-to-right)
subtracted first and then multiplied.
Explain that some calculators carry out operations in the order
they are entered, or left-to-right, in the number sentence. Other
calculators follow a special set of rules that tell which operation to
do first and which operation to do next.
Project 7
936A
Student Page
Introducing Order of
Operations and Computation
Order of Operations
Operations
In many situations, the order in which things are done is important.
For example, you always put on your socks before your shoes.
(Student Reference Book, p. 74H)
In solving number models, certain things must be done in a
certain order. For example, to solve 8 + 4 × 3 = ___, you must
know whether to add first or to multiply first. There are rules
that tell you what to do first and what to do next.
Tell children that to avoid confusion when solving number
sentences, mathematicians have agreed to a set of rules, called the
order of operations. These rules tell you what to do first and
what to do next. Refer children to the list of rules you prepared on
the Class Data Pad.
Rules for the Order of Operations
1. Do operations inside parentheses first.
Follow rules 2 and 3 when computing inside parentheses.
2. Multiply and divide in order, from left to right.
3. Add and subtract in order, from left to right.
Solve. 8 + 4 × 3 = ___
Scientific calculators
always follow the rules
for the order of
operations.
8+4×3
8 + 12
20
Multiply first.
Then add.
The answer is 20.
Rules for the Order of Operations
1. If there are parentheses, do the operations inside the
parentheses first. Follow rules 2 and 3 when computing inside
parentheses.
8 + 4 × 3 = 20
Solve. 10 - (9 - 6 ÷ 2) = ___
Start inside parentheses.
Divide first.
Then subtract.
Subtract again.
The answer is 4.
WHOLE-CLASS
ACTIVITY
10 – (9 – 6 ÷ 2)
10 – (9 – 3)
10 – 6
4
2. Then multiply or divide, in order, from left to right.
10 – (9 – 6 ÷ 2) = 4
3. Finally, add or subtract, in order, from left to right.
74H
74H
NOTE It is important to emphasize that multiplication and division are of equal
Student Reference Book, p. 74H
074A-074J_EMCS_S_G3_SRB_OPE_577260.indd 74H
3/4/11 11:38 AM
priority. This means that you perform whichever of these two operations comes
first, from left to right. Likewise, addition and subtraction are of equal priority.
As a class, read about the order of operations on Student Reference
Book, page 74H.
Practicing Order of Operations
WHOLE-CLASS
ACTIVITY
(Math Masters, p. 394A, Student Reference Book, p. 74H)
Write number sentences on the board and have children copy
them onto their slates. Referring to the Rules for the Order of
Operations on the Class Data Pad or on Student Reference Book,
page 74H, children underline the operation that should be
performed first and then solve. Have volunteers share their
solution strategies. Suggestions:
Project Master
Name
Date
PROJECT
Order of Operations
7
●
Use order of operations to solve each number sentence below. Show your
work. To check your work, use a calculator that follows order of operations.
●
Rules for the Order of Operations
●
1. If there are parentheses, do the operations inside the parentheses first.
Follow rules 2 and 3 when computing inside parentheses.
●
2. Then multiply or divide, in order, from left to right.
3. Finally, add or subtract, in order, from left to right.
1.
2
= 11 – 3 × 3
2. 15 ÷ 3 + 2 =
●
7
●
3.
8
= 20 ÷ 5 × 2
5. (22 + 8) × 2 =
60
4. 6 + 4 ÷ 2 =
6.
16
= (7 × 2) + 14 ÷ 7
Math Masters, p. 394A
936B Project 7 Order of Operations
2
20 = 0 × 20 + 20
12 + 6 ÷ 2 = 15
13 = 15 - 8 + 6
10 ÷ 5 × (2 + 5) = 14
6
= (15 − 6) ÷ 9 + 5
When children are ready, have them work in small groups,
following the rules for the order of operations to solve the number
sentences on Math Masters, page 394A. They use scientific
calculators (those that follow the conventional order of operations)
to check their work. Bring the class together to share answers.
8
386-394A_EMCS_B_MM_G3_PRO_576957.indd 394A
8-2×3=
2/26/11 2:27 PM
2 Extending the Project
Extension Suggestions
PARTNER
ACTIVITY
(Math Masters, p. 394A)
For each problem on Math Masters, page 394A, have children
record the key sequence that should be used to correctly follow the
conventional order of operations with a calculator that follows a
left-to-right order. Children use a left-to-right calculator to check
their work. For example, the key sequence to solve 11 − 3 × 3 is:
3
3
9
11
9
2.
Home Link Suggestions
INDEPENDENT
ACTIVITY
(Student Reference Book, p. 74H)
Children discuss the order of operations and share
Student Reference Book, page 74H with family members.
Children and family members determine whether their
home calculators follow the conventional order of
operations.
Project 7
936C
Name
PROJECT
7
Date
Order of Operations
Use order of operations to solve each number sentence below. Show your
work. To check your work, use a calculator that follows order of operations.
Rules for the Order of Operations
1. If there are parentheses, do the operations inside the parentheses first.
Follow rules 2 and 3 when computing inside parentheses.
2. Then multiply or divide, in order, from left to right.
Copyright © Wright Group/McGraw-Hill
3. Finally, add or subtract, in order, from left to right.
1.
= 11 – 3 × 3
2. 15 ÷ 3 + 2 =
3.
= 20 ÷ 5 × 2
4. 6 + 4 ÷ 2 =
5. (22 + 8) × 2 =
6.
= (7 × 2) + 14 ÷ 7
394A