Number Relations
Order Up
In this activity, students will apply the order of
operations to problems involving whole numbers.
MATERIALS
• Transparency/Page: Code Talkers/Money Exchange
• Transparency/Page: Order of Operations
•Transparency/Page: Order of Operations
Getting Started
• Transparency/Page: Cruise Ship OOPs
• Transparency/Page: PEMDAS
• Transparency/Page: Just for Practice
• Transparency/Page: Using the Order of Operations
• Transparency/Page: Target Number Directions
• blank transparency
VOCABULARY
• order of operations
• exponent
• parentheses
TIME:
Code Talkers/Money Exchange
40 minutes
INTRODUCE
•Display Transparency: Code Talkers/Money Exchange.
(Cover the bottom half of the transparency.)
NUMBER RELATIONS
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Transparency: Code Talkers/Money
Exchange
•Explain, for example, that the Navajo Code Talkers
of World War II confounded our enemies with their
ability to speak in a language that was not only
syntactically different from other languages, but that
also was further encrypted by their use of their own
variants of their native tongue. So successful were
they that even other native Navajo speakers could
not understand what they said. The code was never
broken and was only declassified in 1968.
Number Relations
Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development
Number Relations
•Cover the Code Talkers section of the transparency
and uncover the Money Exchange section.
•Apply the idea of a need for common rules to a
modern setting by discussing the European Union.
Many European countries recently gave up their
own individual currencies and adopted the euro,
a metric-based monetary system that standardized
currency, thus eliminating a variety of systems that
used different rules.
•Explain to students that, fortunately, mathematicians
around the world have agreed to a convention
for basic operations—the order of operations. (If
students ask who defined this, respond that it was an
evolution over time that has become more formalized
since the advent of computers.)
•Emphasize that students need to be conversant with
and competent in the use of the order of operations
because it affects all computation.
•Use a blank transparency to write the following
equation and ask participants for the answer:
3 + 5 • 6 = __. (Answers may include 48 and 33.)
•Ask students which answer is correct. (33) Why?
(Addition is performed after multiplication.)
Order of Operations
First
Parentheses and Exponents
• resolve each individual number
Second
Multiplication and Division
• in order from left-to right
•Ask students what we could do to clarify the
equation. (Use parentheses around the 5 • 6 or
around the 3 + 5.)
•Remind students that whatever is inside parentheses
is done first.
•Display Transparency: Order of Operations and explain.
Third
Addition and Subtraction
• in order from left to right
NUMBER RELATIONS
Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development
Transparency: Order of Operations
Number Relations
Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development
Number Relations
•Do the groupings—anything inside parentheses—
and exponents to resolve individual numbers. Explain
that parentheses and exponents are both used to
group or define elements that should be treated as a
single number. The point is to turn these elements
into single numbers before doing the rest of the
computation.
Order of Operations
Getting Started
400 – 3 • ( 3 + 2 3 )2 + 4 = ____
•Display Transparency: Order of Operations Getting
Started and then write out each mathematical step:
400 – 3 • (3 + 23)2 + 4 = _____
We look at the parentheses and see that before we can
resolve its contents into a single number, we have to
cube 2:
400 – 3 • (3 + 8)2 + 4 = _____
Now, we can add the 3 and the 8 within the
parentheses:
NUMBER RELATIONS
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Transparency: Order of Operations
Getting Started
400 – 3 • (11)2 + 4 = _____
Then, we check for any other exponents and see
that we must still square the 11. Notice that once
the parentheses contain only the single number, the
parentheses can be removed.
400 – 3 • 121 + 4 = _____
Now that all the single numbers have been resolved,
we can continue through the other steps.
Cruise Ship OOPs
•Do multiplication and division from left to right:
12
3
7
1
2
5
4
2
8
9
4
6
510
2
9
400 – 3 • 121 + 4 = _____
400 – 363 + 4 = _____
•Do addition and subtraction from left to right:
3
8
7
1
Order of Departure
Top Deck
k—First
Middle Deck—
k Second
Bottom Deck—
k Third
Board the helicopter left-to-right.
NUMBER RELATIONS
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400 – 363 + 4 = _____
41
37 + 4 = _____
•Display Transparency: Cruise Ship OOPs and suggest
that this is a visual image to help them remember the
order of operations.
Transparency: Cruise Ship OOPs
Number Relations
Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development
Number Relations
PEMDAS
PLEASE EXCUSE MY DEAR AUNT SALLY
CLUE
ACTION
P
is for PARENTHESES.
Perform first.
E
is for EXPONENTS.
Do next.
•Mention PEMDAS, if no one suggests it.
•Display Transparency: PEMDAS.
M
D
A
S
•Ask students if they have seen any other methods for
remembering this information.
is for MULTIPLICATION.
is for DIVISION.
is for ADDITION.
is for SUBTRACTION.-
Perform next,
left to right in order.
Do last,
left to right in order.
NUMBER RELATIONS
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Transparency: PEMDAS
Just for Practice
12 – 24 ÷ 6 + 3 =
4÷2•3=
24 – (4 + 2 • 3) + 5 =
•Indicate that this model, including the verbal
mnemonic (Please Excuse My Dear Aunt Sally), is
used to help them remember the order in which they
should work through problems. It is important for
students to understand the concepts behind the rules.
•Have them look carefully at the PEMDAS model and
then redisplay the problem just completed.
•Ask them if PEMDAS works on this problem.
•Point out, if they do not see, the need to complete
the first exponent before resolving the parentheses.
•Display Transparency: Just for Practice and have
students take out their matching pages.
•Give students 5–6 minutes to work through
the problems.
NUMBER RELATIONS
Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development
Transparency: Just for Practice
Using the Order of Operations
(24 ÷ 2 3 + 2)2 – 4 2 + (32 + 1)3 =
(24 ÷ 8 + 2)2 – 16 + (10)3 =
(3 + 2)2 – 16 + 1,000 =
5 2 – 16 + 1,000 =
9 + 1,000 = 1,009
•Have volunteers come to the front and write their
solutions on the transparency. Answers are shown
below:
◆ 12 – 24 ÷ 6 + 3 = 12 – 4 + 3 = 11
◆ 4 ÷ 2 • 3 = 2 • 3 = 6
◆ 24 – (4 + 2 • 3) + 5 = 24 – 10 + 5 = 19
•Display Transparency: Using the Order of Operations.
(Cover all but the first equation.)
•Give students a minute or two to solve the problem
and then walk through the steps.
•Redisplay Transparency: Order of Operations.
NUMBER RELATIONS
Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development
Transparency: Using the Order of Operations
Number Relations
Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development
Number Relations
•Point out that it is, in most respects, the same as
PEMDAS, except that it groups parentheses and
exponents; it groups multiplication and division
and specifies left to right rather than one before the
other; and it specifies addition and subtraction left
to right as well.
•Point out that PEMDAS is good for remembering
everything that has to be done but that it can be
confusing if one does not understand the underlying
premise. Suggest that if students use PEMDAS, they
must emphasize (1) the grouping of the tasks and
(2) that parentheses and exponents resolve multiple
elements into single numbers before doing the
computation elements.
•Redisplay Transparency: Cruise Ship OOPs.
•Note the order in which the helicopter is evacuating
passengers from the sinking ocean liner. The
evacuation follows the order of operations:
parentheses and exponents
◆
multiplication and division
◆
addition and subtraction
◆
TEACHING TIP: If time permits, ask students to use
PEMDAS rather than simply telling them about it.
Target Number
Directions
The goal of the game is to combine numbers
and symbols to reach a target number or come
as close as you can without exceeding the
number.
DISCUSS AND DO
• The target number is 25.
• Use the numbers 2, 3, 6, and 9.
• Each number must be used and can be used
only once.
• The numbers can be used in any order.
• You may use only one of the numbers as an
exponent.
• Operation symbols (i.e., +, –, •, ÷) and
parentheses may be used multiple times
NUMBER RELATIONS
Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development
Transparency: Target Number Directions
•Tell students that they now will apply the order
of operations.
•Display Transparency: Target Number Directions.
•Explain that the goal of the game is to combine
numbers and symbols to reach a target number or
come as close as possible without exceeding
the number.
Number Relations
Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development
Number Relations
•Review the directions:
Set a target number of 25. (25 is an arbitrary
number. In the classroom, students may choose
in the same manner or use number cubes or other
methods to set a target.)
◆
Use the numbers 2, 3, 6, and 9 for your example
and one practice round. Then, erase these and write
four other random numbers.
◆
Each number must be used and can be used only
once.
◆
The numbers can be used in any order.
◆
You may use only one of the numbers as an
exponent.
◆
Operation symbols (i.e., parentheses, +, –, •, ÷) can
be used multiple times and in any order.
◆
•Show the example below for a target of 25:
3 • 6 + 9 – 2 = 25
18 + 9 – 2 = 25
27 – 2 = 25
•Show a second possible equation if students do not
mention it.
(9 – 6)3 – 2 = 25
•Set a new target number and have students create
their own equations.
•Give students a couple of minutes to work on their
equations and then ask if anyone has hit the target.
•Have the student who came closest to the target
number (without going over) come to the front and
write on a blank transparency the steps that he or
she used.
•Repeat the activity a second time using different
starting numbers.
Number Relations
Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development
Number Relations
TEACHING TIP: If students have used parentheses
in their equations, examine these equations with
students to determine if the parentheses were
necessary or if the order of operations alone would
have suggested the same answers.
CONCLUDE
Review with students how using order of operations
helped them in the game. They should realize
that if they wanted to add or subtract a quantity
before multiplying or dividing, they needed to use
parentheses.
End of Order Up
Number Relations
Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development