Order of Operations
A Fairly Ordered Operation
SUGGESTED LEARNING STRATEGIES: Close Reading, Think
Aloud, Marking the Text, Summarize/Paraphrase/Retell,
Quickwrite
ACTIVITY
ACTIVITY 2.9 Investigative
2.9
Order of Operations
Activity Focus
My Notes
• Order of Operations
Ayana and Zachary Wilson are excited about going to the Pace
County Fair. General admission to the fair is $8.00 per person.
It covers visiting exhibits and some entertainment. Tickets for
rides and games must be bought separately. Food and drinks are
purchased at the concession stands. A ride ticket costs $3.00.
Materials
• No special materials are needed.
Chunking the Activity
1. Ayana loves to make lists of things to do to prepare for an
activity. She made the following list for the morning of the fair.
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To Do
#1
#2–4
#5
#6–7
Order
Comb my hair
8
Take a shower
2
Brush my teeth
1
Eat breakfast
6
Put my clothes on
3
Buy my tickets for the fair
10
Put on my shoes
5
Put on my socks
4
Ride to the fair
9
Get money out of piggy bank
7
#12
#13
#14
#15
Paragraph Close Reading,
Think Aloud, Marking the Text,
Summarize/Paraphrase/Retell
1 Quickwrite After the students
give the order of the steps it is
very important that they explain
why the order of the steps is
important.
Y
You could have students
w
write each step on a
sticky note or an index card so
they can easily play around with
the order of the steps before
entering the order in the student
book.
TEACHER TO
TEACHER
a. Order the steps as you think Ayana will complete them.
Answers may vary. Sample answer: See table above.
b. Explain why the order of steps is important.
Answers may vary. Sample answer: Some things have to be
done before others, such as putting on socks before shoes.
2. Ayana plans to go on 5 rides. She wrote the numerical
expression 8 + 5 × 3 to represent the cost of her rides and
admission to the fair.
2 Close Reading, Guess and
a. What is the total cost of Ayana’s admission to the fair and the
rides she wishes to go on?
$23
b. Do you think her expression represents this total cost?
Explain.
Answers may vary. Sample answer: Yes, if you multiply first.
Unit 2 • Operations with Numbers
115
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Check, Quickwrite The intent of
this question (and the next two) is
to show a need for an agreement
on the order in which operations
should be performed. After completing this first set of questions,
students will learn that multiplication should be performed before
addition.
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115-122_SB_MS1_2-9_SE.indd 115
#8
#9
#10
#11
Unit 2 • Operations with Numbers
115
ACTIVITY 2.9 Continued
ACTIVITY 2.9
continued
Order of Operations
A Fairly Ordered Operation
3 Close Reading, Guess and
Check, Quickwrite
SUGGESTED LEARNING STRATEGIES: Close Reading, Guess
and Check, Quickwrite, Work Backward
My Notes
4 Debrief Students should
3. Zachary intends on going on 8 rides in the morning and
5 rides in the afternoon. He wrote the expression 8 + 5 × 3
to represent the total cost of the rides he wishes to go on.
determine that multiplication
should come before addition.
Students should also determine
that the way expressions are
written is important.
a. What is the total cost of Zachary’s rides?
$39
b. Do you think his expression represents this total cost?
Explain.
5 Close Reading, Guess and
Check, Quickwrite Students
should be able to explain that
multiplication is performed before
addition.
Answers may vary. Sample answer: Yes, if you add first.
4. Is it a problem that both Ayana and Zachary used the same
mathematical expression to represent two different costs?
Explain.
Answers may vary. Sample answer: Yes, because the
expressions have different values and it is not clear which
is correct.
6 Close Reading, Guess and
Check, Quickwrite With this
question, students’ understanding
of order of operations for adding
and subtracting continues to be
developed.
Mathematicians have agreed that when evaluating an expression
containing both addition and multiplication the operation of
multiplication should be performed first.
5. Can Ayana use the expression 8 + 5 × 3 to represent the cost of
her admission ticket and rides or can Zachary use it to represent
the cost of his rides? Explain.
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Ayana, because multiplication must be performed before
addition.
The day of the fair both Ayana and Zachary realized that they had
not figured food into the cost of going to the fair.
6. Ayana has $100 in her piggy bank. She took $60 out to go to the
fair. Then her dad gave her $5 allowance. She does not need this
$5 for the fair so she puts it in her piggy bank. Ayana wrote the
expression 100 - 60 + 5 to represent the amount of money that
is now left in her bank.
a. What is the total amount left in Ayana’s piggy bank?
$45
b. Do you think her expression represents this total amount?
Explain.
Answers may vary. Sample answer: Yes, you can add or
subtract first and the answer is the same.
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Order of Operations
ACTIVITY 2.9
A Fairly Ordered Operation
SUGGESTED LEARNING STRATEGIES: Close Reading,
Guess and Check, Work Backward, Quickwrite, Think
Aloud, Marking the Text, Summarize/Paraphrase/Retell
ACTIVITY 2.9 Continued
continued
7 Guess and Check, Quickwrite Some students may see
that enclosing 60 + 5 in parentheses and then subtracting from 100
makes the expression true.
My Notes
7. Zachary has $100 in his piggy bank. He took $60 out for rides,
food, and admission to the fair, and he took $5 out for a souvenir.
Zachary wrote the expression 100 - 60 + 5 to represent the
amount of money now left in his bank.
A
After Question 7 would
b a good time to have
be
c
students connect
mathematical
properties and order of operations
to the rules by which they play a
game, such as soccer, chess, or
volleyball.
TEACHER TO
TEACHER
a. What is the total amount left in Zachary’s piggy bank?
$35
b. Do you think his expression represents this total amount?
Explain.
Answers may vary. Sample answer: No, because you should
not add when you take something away.
Think about the order in which operations should be performed.
When evaluating a numerical expression with addition and
subtraction, the operations of addition and subtraction should be
performed in the order in which they appear from left to right.
Paragraph Close Reading
8 Guess and Check, Quickwrite Students will use what they
just learned about the order in
which to do addition and subtraction to do this problem.
8. Does the expression 100 - 60 + 5 represent the amount left in
Ayana’s bank or the amount left in Zachary’s bank? Explain.
It represents the amount in Ayana’s bank: 100 - 60 + 5 = 45.
After that Zachary decided to take only $60 to the fair.
© 2010 College Board. All rights reserved.
Ayana and Zachary’s mom decided they needed to take some
snacks to the fair. She took a bag of 6 granola bars and divided
the bars evenly into three sacks. She did the same thing with three
more bags of 6 granola bars, sharing them evenly into the same
three sacks. She then gave Ayana and Zachary each a sack.
Paragraph Close Reading,
Summarize/Paraphrase/Retell
9 This question gives students
practice in deciding the order in
which to perform multiplication
and division.
9. When Ayana asked Zachary how many granola bars were in
each sack, he said the expression 6 ÷ 3 × 4 represented the
number of granola bars.
a. Ayana evaluated the expression and said the bag contained
less than 1 whole granola bar. Explain how she arrived at this
answer.
Answers may vary. Sample answer: 6 ÷ 3 × 4 = 6 ÷ 12 = .5.
b. How many granola bars were actually in each bag? Explain
how Zachary derived his expression.
Explanations may vary. Sample answer: 8 bars;
6 bars ÷ 3 = 2; 2 × 4 bags of bars = 8.
Unit 2 • Operations with Numbers
12/16/09 5:57:12 PM
© 2010 College Board. All rights reserved.
PM
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117
Unit 2 • Operations with Numbers
117
ACTIVITY 2.9
continued
Paragraph Close Reading,
Think Aloud, Marking the Text,
Summarize/Paraphrase/Retell
0 Guess and Check, Create
Representations This is a summative question that asks students
to use their knowledge of the
order of operations for addition,
subtraction, multiplication and
division to play the game. Point
out that the row of three can be
horizontal, vertical, or diagonal.
To extend this activity, you might
have students make their own
game cards on 3 × 3 grids,
putting the numbers from 1
through 9 in any order they
choose. Students may want to try
using numbers other than 2 with
the number sentences.
Suggested Assignment
CHECK YOUR UNDERSTANDING
p. 122, #1–6
Order of Operations
A Fairly Ordered Operation
SUGGESTED LEARNING STRATEGIES: Close Reading, Think
Aloud, Marking the Text, Summarize/Paraphrase/Retell,
Guess and Check, Create Representations
My Notes
ACADEMIC VOCABULARY
The order of operations is
a set of rules for evaluating
expressions with more than one
operation. The order is:
• Do calculations inside
parentheses first.
• Evaluate expressions with
exponents.
• Multiply or divide from left
to right.
• Add or subtract from left
to right.
Tic-tack-two game card
Numerical expressions can contain the operations of addition,
subtraction, multiplication, and division, so it is important to
follow an order of operations to avoid confusion. The operations
of multiplication and division are evaluated from left to right. The
operations of addition and subtraction are also evaluated from left
to right.
10. At the fair Ayana used one ticket to play Tic-Tack-Two. She
was handed a Tic-Tack-Two game card and this score sheet.
To play, each person should fill in the spaces between the 2s
with any operation sign: +, -, ×, or ÷ to equal the number at
the end of the row. You can use an operation more than once in
a row. When you get a number, put an X over that number on
your Tic-Tack-Two card. The first player to get three in a row
wins. An example, which is not a value on the game card, has
been done for you.
Play this game with the members of your group. Remember to
use the order of operations.
8
1
6
3
5
7
2
÷
2
+
2
÷
2
-
2
=
0
4
9
2
2
×
2
-
2
÷
2
-
2
=
1
2
+
2
+
2
-
2
-
2
=
2
2
+
2
-
2
+
2
÷
2
=
3
2
÷
2
+
2
÷
2
+
2
=
4
2
+
2
-
2
÷
2
+
2
=
5
2
+
2
+
2
+
2
-
2
=
6
2
×
2
×
2
-
2
÷
2
=
7
2
×
2
×
2
×
2
÷
2
=
8
2
×
2
×
2
+
2
÷
2
=
9
Answers may vary. Sample
answers shown.
UNIT 2 PRACTICE
p. 137, #59–63
© 2010 College Board. All rights reserved.
ACTIVITY 2.9 Continued
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118 SpringBoard® Mathematics with Meaning™ Level 1
Order of Operations
ACTIVITY 2.9
A Fairly Ordered Operation
ACTIVITY 2.9 Continued
continued
SUGGESTED LEARNING STRATEGIES: Close Reading,
Think Aloud, Marking the Text, Summarize/Paraphrase/
Retell, Guess and Check, Work Backward, Quickwrite
a Close Reading, Think Aloud,
Marking the Text, Summarize/
Paraphrase/Retell, Guess and
Check, Work Backward, Quickwrite (b, c) The intent of this
question is to show that exponents must be evaluated before
multiplication.
My Notes
11. Zachary decided to join Ayana in a game of darts. They found
the following target to be very unusual.
33
32
3
Each player gets to throw two darts at the target. The numbers
on the target are the points a player gets for each dart that lands
in that region of the target. Both of Zachary’s darts landed in
the gray region of the target.
a. Ayana expressed his score as 32 + 32. What did Ayana get for
his Zachary’s score?
18
© 2010 College Board. All rights reserved.
b. Zachary expressed his score as 2 × 32. Explain how he would
have to evaluate this expression so that he would get the
same score as Ayana did?
Answers may vary. Sample answer: He would have to
evaluate 32 before he multiplies.
c. What do you think is the order that mathematicians have
agreed upon when evaluating an expression with exponents
and multiplication?
Exponents must be evaluated before multiplication.
Unit 2 • Operations with Numbers
12/16/09 5:57:18 PM
© 2010 College Board. All rights reserved.
PM
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119
Unit 2 • Operations with Numbers
119
ACTIVITY 2.9 Continued
Paragraph Close Reading,
Think Aloud, Marking the Text,
Summarize/Paraphrase/Retell
ACTIVITY 2.9
continued
Order of Operations
A Fairly Ordered Operation
My Notes
SUGGESTED LEARNING STRATEGIES: Close Reading,
Think Aloud, Marking the Text, Summarize/Paraphrase/
Retell, Guess and Check, Work Backward
As they drove home from the fair, Zachary wondered if there was
a way to change the order in which an expression is evaluated.
T heir father explained that parentheses can be placed around the
numbers and operations to be performed first.
b Guess and Check, Work
Backward The intent of this
question is to show how parentheses can change the order in
which operations are performed.
12. Add parentheses to the expression 8 + 5 × 3 so that it can be
evaluated in such a way as to give Zachary the cost of buying
tickets for 8 rides in the morning and 5 rides in the afternoon at
$3.00 per ride.
Paragraph Close Reading,
Think Aloud, Marking the Text,
Summarize/Paraphrase/Retell
(8 + 5) × 3
Mathematicians have agreed that in evaluating an expression
involving grouping symbols, such as parentheses or brackets, the
operations inside the grouping symbols should be completed before
completing those operations outside the grouping symbols.
c Guess and Check, Work
Backward This question provides
practice evaluating an expression
with parentheses.
13. T he Wilsons decide to stop at a restaurant on the way home
from the fair. Part of the menu is shown below.
© 2010 College Board. All rights reserved.
SNACKS
G`qqX%%%%%%%%%%%%%%%%%%%%%%%% *%''
JXcX[%%%%%%%%%%%%%%%%%%%%%%%% ,%''
Al`Z\%%%%%%%%%%%%%%%%%%%%%%%% )%''
8ggc\j%%%%%%%%%%%%%%%%%%%%%%% )%''
Pf^lik%%%%%%%%%%%%%%%%%%%%%% )%,'
?fdX[\Jflg%%%%%%%%%%%%%%%% +%''
VALUE MEALS
G`qqX#Pf^lik#Al`Z\%%%%%%%%%%%% -%''
?XdYli^\i#Jflg#Al`Z\%%%%%%%% /%''
The Wilsons order one hamburger value meal, four apples, two
juices, and two pizza value meals.
a. Write an expression to find the total cost of the family’s meal.
8 + (4 + 2) × 2 + 2 × 6
b. Find the total cost of the family’s meal. Show your work.
$32
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120 SpringBoard® Mathematics with Meaning™ Level 1
Order of Operations
ACTIVITY 2.9
A Fairly Ordered Operation
ACTIVITY 2.9 Continued
continued
d This is a summative question
SUGGESTED LEARNING STRATEGIES: Summarize/
Paraphrase/Retell, Create Representations
that asks students to put all of the
facts they have learned together
and come up with a set of steps
describing the order in which
a numerical expression can be
evaluated.
My Notes
14. Since every numerical expression should have only one value,
mathematicians have agreed upon an order of operations for
evaluating expressions. Describe the steps involved in simplifying
a numerical expression using order of operations.
Step 1: Evaluate all expressions in parentheses.
Step 2: Evaluate all expressions involving exponents.
Paragraph Close Reading,
Think Aloud, Marking the Text,
Summarize/Paraphrase/Retell
Step 3: Complete all multiplication and division in order from
left to right.
Step 4: Complete all addition and subtraction in order from left
to right.
When Ayana arrived home she wanted to f igure out how much she
and Zachary had left of the money they took to the fair.
• Both she and Zachary paid $8 for an admission ticket.
• She went on a total of 5 rides and Zachary went on 8 rides in
the morning and 5 rides in the af ternoon at $3 per ride.
• She played 2 games and he played 1 at $3 per game.
• Zachary bought a $5 souvenir.
The following expression represents the total amount of money
she and Zachary have lef t of the $60 that they each took with them
to the fair.
60 × 2 - [2 × 8 + ((8 + 5) + 5) × 3 + (1 + 2) × 3 + 5]
© 2010 College Board. All rights reserved.
Notice the parentheses within parentheses. T hese are
called nested parentheses. You must always work the innermost
parentheses first. Notice too that brackets have been used. Brackets
can be used in place of parentheses, usually on the outside of
parentheses.
Unit 2 • Operations with Numbers
12/16/09 5:57:24 PM
© 2010 College Board. All rights reserved.
PM
115-122_SB_MS1_2-9_SE.indd 121
121
Unit 2 • Operations with Numbers
121
ACTIVITY 2.9 Continued
ACTIVITY 2.9
continued
e Summarize/Paraphrase/
Retell, Create Representations
This question provides practice
using nested parentheses.
Order of Operations
A Fairly Ordered Operation
SUGGESTED LEARNING STRATEGIES: Close Reading,
Think Aloud, Marking the Text, Summarize/Paraphrase/
Retell, Guess and Check, Work Backward
My Notes
15. Ayana started to evaluate the expression. Finish her work.
Notice that parentheses around a single number can be dropped
without changing any values.
Suggested Assignment
CHECK YOUR UNDERSTANDING
p. 122, #7–12
60 × 2 - [2 × 8 + ((13) + 5) × 3 + (1 + 2) × 3 + 5]
UNIT 2 PRACTICE
p. 137, #64
60 × 2 - [2 × 8 + 18 × 3 + 3 × 3 + 5]
60 × 2 - [2 × 8 + ((8 + 5) + 5) × 3 + (1 + 2) × 3 + 5]
60 × 2 - [2 × 8 + (18) × 3 + (3) × 3 + 5]
60 × 2 - [16 + 54 + 9 + 5]
60 × 2 - [16 + 54 + 9 + 5]
2. 40
3. 5
CHECK YOUR UNDERSTANDING
4. 80
Write your answers on notebook paper.
Show your work
Insert parentheses when needed to make the
number sentence true.
Simplify using order of operations.
7. 11 + 8 × 4 = 43
1. 18 - 12 ÷ 2 × 3
8. 5 × 2 + 3 = 25
5. 8
6. 17
7. No parentheses are needed.
2. 9 × 4 + 8 ÷ 2
9. 16 × 4 - 4 × 4 = 0
8. 5 × (2 + 3) = 25
3. 2 × (8 + 2) ÷ 4
9. 16 × (4 - 4) × 4 = 0
4. (1 + 3)2 × 5
Use parentheses and the symbols +, -, ×, and
÷ to make each number sentence true.
10. Answers may vary. Sample
answer: 7 + (3 - 1) × 1 = 9
5. 4 × 42 ÷ (56 ÷ 8 × 3)
2
6. 4 × 2 + 1
© 2010 College Board. All rights reserved.
60 × 2 - 84
120 - 84
36
1. 0
10. 7 ____ 3 ____1 ____1 = 9
11. 10 ____ 5 ____ 5 ____ 2 =10
12. MATHEMATICAL Give some reasons why
R E F L E C T I O N mathematicians had to
agree on an order of operations to be used
to simplify a numerical expression.
11. Answers may vary. Sample
answer: 10 + (5 - 5) × 2
12. Answers may vary. Sample
answer: Situations such as
those in this lesson can result
in confusion when writing
an expression unless there is
understanding about how to
write expressions and how to
interpret them.
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122 SpringBoard® Mathematics with Meaning™ Level 1