Order
of
Operations
Unit Description
Susan Mercer
The NCTM standards contend that “... learning mathematics without understanding has
long been a common outcome of school mathematics instruction.” (page 20 NCTM standards) In
particular this is true for order of operations. Traditionally, textbooks have presented order of
operations as a convention, a cluster of rules to be followed and memorized in order to get the right
answer. For example, Mc Dougal Littell states, in the California Middle School Mathematics:
Concepts and Skills Course 1 textbook, “To make sure everyone gets the same result when an
expression is evaluated, mathematicians have established order of operations.” These rules do not
provide understanding; nor do they build on students’ prior knowledge of numbers and operations.
Students are expected to apply the rules without comprehending why they are doing what they are
doing. Furthermore, the NCTM standards argue that “Students who memorize facts of procedures
without understanding often are not sure when or how to use what they know, and such learning is
often quite fragile.” (page 20 NCTM standards) Mathematical expressions are presented to
students out of context, without any meaning attached to the numbers or operations. This leads to
mistakes primarily because students tend to solve mathematical expression from left to right
rather than looking at the problem as a whole. In my experience, students view mathematical
expressions as abstract and meaningless and tend to “decode” them in the same way as they
decode a sentence or paragraph, from left to right.
Also, many teachers encourage students to use the acronym “Please Excuse My Dear Aunt
Sally” (PEMDAS) to remember the order in which to evaluate mathematical expressions.
Unfortunately, PEMDAS does not provide an explanation and students tend to solve the
multiplication before the division rather than multiplication and division from left to right and
addition before subtraction rather than addition and subtraction from left to right.
The order of operation model described in this article requires students to represent
mathematical expressions in three different ways: 1) verbally; 2) using a drawing; and 3)
symbolically using addition and/or subtraction only. These representations help students “see” the
mathematical expressions and help them evaluate them by finding the number of objects in the
representation. By using this model students “…gain access to mathematical representations and
the ideas they represent, they have a set of tools that significantly expand their capacity to think
mathematically.” (page 37 NCTM standards) Modeling order of operations using different
representations helps students comprehend how to evaluate a mathematical expression by
making sense of the parts of the expression rather than applying memorized rules. In addition,
scaffolding methodologies are used throughout. The unit starts by reviewing what students
already know and build on that knowledge. Each new concept or idea is introduced one at a time
and layered upon the concepts already presented.
© S. Mercer - Order of Operations Description
page - 2
This model was developed on addition and multiplication seeing that students learned these
two concepts in the primary grades. Addition is the operation of combining two or more elements
or groups of elements and multiplication denoting successive additions. For example: three times
seven is the same as three groups of seven and this is the same as 7 + 7 + 7. Further, exponents
can be expressed as successive multiplications and multiplications can be expressed as successive
additions. For example: three to the power of two is the same as three times three, which is the
same as three groups of three which can be written as three plus three plus three.
Mathematically: 32 = 3• 3 = 3 + 3 + 3.
Expression
A graphic organizer is used to help students appreciate the different representations of a
mathematical expression. The above example presents the graphic organizer. In the middle oval
students write the mathematical expression. In the top left box, students record the expression in
words as they “read” it from left to right. In the top right box, students represent the expression
using individual objects and/or groups of objects. In the bottom left box, students use the pictorial
representation to rewrite the mathematical expression using only addition and/or subtraction. In
the final box, students write the answer after evaluating the mathematical expression.
When using this model, it is very important not to provide students with the order of
operation rules as they are stated in the textbook. The hierarchies between the different
operations need to be “discovered” and generalized by the students as they move through the unit.
The teacher is a key element in guiding and facilitating student discussion in order for them to
make the needed connections and generalizations. It is crucial for the teacher to model how to
complete the graphic organizer. The teacher should model each example by asking probing
questions and recording the students’ answers on the graphic organizer.
© S. Mercer - Order of Operations Description
page - 3
The teacher introduces the graphic organizer by reviewing multiplication with a simple
problem such as three times seven. The above example shows the completed graphic organizer for
the problem three times seven. In the top left box, students represent the problem using words
“three groups of seven”; in the top right box students draw three circles to represent three groups
and inside each group they draw seven objects; in the bottom left box the students represent the
multiplication problem as successive additions, in this case seven plus seven plus seven. Students
might not remember or know the concept of multiplication as successive additions therefore it is
important for them to complete the bottom left box by observing their picture. Students need to
know that to evaluate an expression means, in this context, to find out how many objects they
have in total. This is easily accomplished by looking at the picture. The bottom right box shows
the total number of objects. If students are familiar with the representation of multiplication as
“groups of“ and successive additions, then they should evaluate one or two more multiplication
problems. If this is a new concept for them then more problems are necessary.
Next, an addition operation is added to the multiplication problem. It is crucial that the first
example shows the addition first when reading the problem from left to right because in this way
students do not make the erroneous assumption that expressions are evaluated as they are read
from left to right. It is very important for teachers to model on the board or overhead how to
complete the graphic organizer and model the different expressions, especially when a new
component or level of difficulty is added to the previous problems.
Expression
The above example displays a complete graphic organizer for the expression five plus two
times three. It is essential for the teacher to help students make connections between the
different representations and the idea that even though the five is first when the expression is read
from left to right, mathematically the number of objects in the groups is evaluated first.
© S. Mercer - Order of Operations Description
page - 4
In words
Representation
Emphasis should be placed
on the number of objects.
three
plus
two groups of eight
plus
two
3+2•8+2
Expression
Can you think
of a shortcut
to solve this
problems?
3+8+8+2
21
Expression with addition only
Answer
In words
Representation
Emphasis should be placed
on the number of objects.
two groups of six
plus
four groups of four
2•6+4•4
Expression
Can you think
of a shortcut
to solve this
problems?
6+6+4+4+4+4
28
Expression with addition only
Answer
The above examples demonstrate the completed graphic organizer for different expressions
that include addition and multiplication. At this point it is important to challenge the students to
find “shortcuts”. For example, some students may ask “can’t we just write the number, instead of
drawing all the pictures”. Or they may suggest, writing directly twelve instead of “6 + 6” or writing
“16” instead of writing “4 + 4 + 4 + 4”. It is crucial for the teacher to challenge students to explain
why they are able to write their shortcuts and where in the different boxes or representations they
can find the short cut. For example: “6 + 6” is two groups of six in the verbal representation, it is 2
• 6 in the expression box, it is the two groups of six in the drawing and it is “6 + 6” in the expression
with addition only box. It is key to have students note that they are not really solving the problem
from left to right when using their shortcut but rather looking at the problem as a whole and
finding the “groups” to find out how many objects are in each group.
It is critical for the teacher to constantly ask probing questions such as: How many groups
do you have? How many objects are in each group? What part of the expression represents the
number of groups? What part of the expression represents the number of objects in each group?
What is outside of the groups? Are all the groups the same? Why? Why not?
© S. Mercer - Order of Operations Description
page - 5
Once students are able to represent and evaluate expressions that involve addition and
multiplication, parenthesis is incorporated into the expression. This is a new type of problem and
the teacher needs to model how to solve the first expression with the students, while asking
probing questions such as: How is this problem the same as the ones we did before? How is it
different? It is very important for students to make connections with what they learned previously
and realize that the logic that they applied before is still valid with the new concept.
Different shapes or colors can be used to represent what is inside each group but it is
important that students focus on the total number of objects in each group and not their shape or
color. Also, the teacher needs to ask students: What is inside the group? How many groups are
there?
4 (2 + 3)
Expression
Expression
The above examples display the graphic organizer completed for two problems that include
parenthesis. Next, students solve different problems that include addition, multiplication and
parenthesis.
© S. Mercer - Order of Operations Description
page - 6
In words
Representation
Emphasis should be placed
on the number of objects.
three groups of
two groups of four
3•2•4
Expression
Can you think
of a shortcut
to solve this
problems?
4+4 + 4+4 +4+4
24
Expression with addition only
Answer
Subsequently, students apply what they have learned to problems involving successive
multiplications. The above example presents the expression three times two times four. The
picture represents three groups, each one containing two groups of four elements. During the
discussion time students are able to describe “shortcuts” such as six groups of four or three groups
of eight. This is an appropriate time to discuss the commutative and associative properties.
Next, students are presented with expressions including exponents. Students are introduced
to this concept with the expression three to the power of two. In words three to the power of two is
three times three, which is represented as three groups of three. Using an expression with addition
only, three to the power of two is three plus three plus three. The answer is nine. After modeling
one or two examples with exponents only, exponents and addition is presented to students. The
graphic organizer below shows a completed graphic organizer for the expressions using addition
and exponents.
4 2+ 3
Expression
Once students are familiar with the different types of expressions that involve addition,
multiplication, parenthesis and power of two, subtraction is introduced to the model.
© S. Mercer - Order of Operations Description
page - 7
In words
Representation
ten minus nine
10 - 9
Problem
10 – 9
1
Expression with addition
and/or subtraction only
Answer
First students need to review the concept of subtraction as taking away. A simple
expression such as 10 minus 9 can be used as review. Students complete the graphic organizer for
subtraction. It is critical for the teacher to show students what notation will be used to represent
take away in order for all students to use the same visual aid.
Next, the teacher presents an expression such as 10 – 3 • 2. In words students write ten
minus three groups of two. When representing this expression students need to draw ten objects
and then take away three groups of two. Mathematically using addition and/or subtraction
students need to record 10 – 2 – 2 – 2. Ten represents the starting number of objects and two is
the number of objects the students need to take away three times in order to represent the
expression accurately.
In words
Representation
take away
four groups of four
from
six groups of three
6•3-4•4
Expression
3+3+3+3+3+3–
4–4–4–4
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations Description
Can you think
of a shortcut
to solve this
problems?
2
Answer
page - 8
In words
Representation
four groups
of
nine minus five
4 ( 9 - 5)
Expression
9–5 + 9–5 + 9–5 + 9–5
16
Expression with addition and/or
subtraction only.
Answer
The above example shows a completed graphic organizer that combines subtraction and
multiplication and the above example llustrates an expression that combines parenthesis and
subtraction. It is imperative to introduce each type of situation individually and model it for the
students. In this way students are able to “see” how to represent each expression and can
conceptually understand what is being done to solve each expression. Due to space constrains not
all the different examples can be presented.
In words
Representation
Sabrina has six boxes of crayons,
with eight crayons in each box. She
uses ten crayons. How many
crayons does she have left?
6 • 8 – 10
Expression
38 crayons
8+8+8+8+8+8 – 10
Expression with addition and/or
subtraction only.
Answer
The graphic organizer can also be used to solve word problems. Using this model helps
students write expressions given a word problem. The above example illustrates a word problem
where students are asked not only to solve the problem but also to write a mathematical
expression to represent the problem. Representing problems visually helps students see the
groups and the objects that are added that are not in groups.
© S. Mercer - Order of Operations Description
page - 9
Once students are familiar with the different types of problems and their representations, it
is important for students to work in groups to discuss, draw conclusions and summarize in what
order expression are evaluated. To help this process students are asked to complete a summary
table that has three columns.
Evaluating Expressions
with the Representation
Evaluating Expressions
without the Representations
10 + 3 • 4
10 + 3 • 4
ten
plus
three groups of four
10 + 12
4
10
4
22
Order the expression
was evaluated
1) multiplication
2) addition
4
10 + 4 + 4 + 4 = 22
5
5 + 3 ( 2 + 1)
5 + 3 ( 2 + 1)
five
plus
three groups of two plus one
5+3•3
2+1
2+1
5+9
1) parenthesis
2) multiplication
3) addition
14
2+1
5 + 3 + 3 + 3 = 14
The above table illustrates this process. In the first column students evaluate an
expression using a picture. Next, they are asked: “How would you evaluate this expression without
the representation?” It has been my experience that many students when asked this question
revert to looking at the problem from left to right. But as soon as they realize that the answer they
get from evaluating the expression from left to right does not coincide with the answer they arrived
at by drawing the picture they self-correct. The graphic organizer provides students with
immediate feedback; they can check their answers by looking at the picture. Lastly students are
asked: “While evaluating the expression without the picture, what operation did you do first?
Why? What did you do next?” Students record in the last column the order they evaluated the
expression. After several examples, a pattern emerges that enables students to generalize the
order of operation rules. These generalizations can then be extended to expressions that are
difficult to represent with a picture such as expressions with fractions, decimals, and division.
At this point it is necessary to review with the students the importance of evaluating an
expression from left to right if it includes only addition and subtraction. This can be done by writing
the following expression on the board: 10 – 4 + 3 and asking students to represent it using words
and a picture. Students will write “ten minus four plus three” and will represent the expression by
drawing ten objects, taking away four and finally adding three leaving nine objects. Next, ask
students: “Do you get the same answer if you add four and three and then subtract the answer
from ten? Why not?” Students quickly realize that if they add first they are subtracting seven
from ten instead of subtracting four then adding three. Using several of these examples can be
© S. Mercer - Order of Operations Description
page - 10
used to reinforce and emphasize the importance of evaluating an expression from left to right if it
only includes addition and subtraction. After students are familiar with these types of problems,
the same type of activity can be used with multiplication and division.
Probing questions such as: How can you explain to a friend how to solve a problem if you
cannot draw a picture? Do you always solve problems from left to right? Is there an order in which
problems are solved? Is there a hierarchy? Students should be able to articulate the conclusion
they have found regarding order of operations. In my experience each student can arrive at two or
three conclusions, such as all multiplications can be expressed as additions but not vice versa.
Consequently a class discussion where all the conclusions are presented, recorded and summarized
is critical. This dialogue allows students to thoroughly understand how to evaluate mathematical
expressions. By the end of this unit students should be able to answer the following key questions:
Why are addition and subtraction done last? What are parentheses used for in an expression?
Why were the rules of order of operations established? Why were they selected? Are the rules
arbitrary or is there a reason for them?
Extension
Using the same reasoning process, this model can be used with variables to introduce the
distributive property. The first expression should be “3a” represented as three groups of “a” which
is the same as “a + a + a”. The graphic organizer below shows how this model can be used to teach
distributive property by changing only the last box from “Answer” to “Expression without
Parenthesis.” It is crucial for the teacher to continually ask students: How many groups are
there? What is in each group? What part of the expression is not part of the group? What part of
the expression denotes the number of groups? What part of the expression indicates the number
of objects in each group?
In words
Representation
five plus
two groups of
a plus three b plus four
5
a
b b
b
4
a
b b
b
4
5 + 2 ( a + 3b + 4 )
Expression
5+
a+b+b+b+4
+a+b+b+b+4
Expression with addition only
© S. Mercer - Order of Operations Description
2a + 6b + 13
Expression without parenthesis
page - 11
Conclusion
This model is not magical. It requires the teacher to plan and prepare the examples
carefully and to continually ask probing questions that help students make the connections
between the mathematical expression and its representations. Furthermore, the teacher needs to
assess the needs of his/her students and determine the pace and number of examples each type of
problem requires before introducing the next level difficulty. Is the time and effort worthwhile?
Absolutely! By using this unit students deepen their understanding of number and operation and
improve their number sense
© S. Mercer - Order of Operations Description
page - 12
Order
of
Operations
Answer Key
Susan Mercer
Multiplication
In words
Representation
Emphasis should be placed
on the number of objects.
three groups of seven
3 • 7
Expression
21
7+7+7
Expression with addition only
In words
Answer
Representation
Emphasis should be placed
on the number of objects.
four groups of five
4 • 5
Expression
20
5+5+ 5+5
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
page - 1
In words
Representation
Emphasis should be placed
on the number of objects.
three groups of three
3 • 3
Expression
9
3+3+3
Expression with addition only
In words
Answer
Representation
Emphasis should be placed
on the number of objects.
one group of nine
1 • 9
Expression
9
9
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
page - 2
Multiplication and Addition
In words
Representation
Emphasis should be placed
on the number of objects.
five
plus
two groups of three
5+2 • 3
Expression
11
5+3+3
Expression with addition only
Answer
In words
Representation
Emphasis should be placed
on the number of objects.
nine
plus
three groups of four
9 + 3 • 4
Expression
21
9+4+4+4
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
page - 3
Emphasis should be placed
on the number of objects.
In words
Representation
seven groups of four
plus
two
7•4+2
Expression
Can you think
of a shortcut
to evaluate this
expression?
4+ 4+ 4+ 4+ 4+ 4 +4 +2
30
Expression with addition only
Answer
In words
Representation
Emphasis should be placed
on the number of objects.
ten
plus
six groups of two
10 + 6 • 2
Expression
Can you think
of a shortcut
to evaluate this
expression?
10+ 2+ 2+ 2+ 2+ 2 +2
22
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
page - 4
In words
Representation
Emphasis should be placed
on the number of objects.
three
plus
two groups of eight
plus
two
3+2•8+2
Expression
Can you think
of a shortcut
to evaluate this
expression?
3+8+8+2
21
Expression with addition only
Answer
In words
Representation
Emphasis should be placed
on the number of objects.
two groups of six
plus
four groups of four
2•6+4•4
Expression
Can you think
of a shortcut
to evaluate this
expression?
6+6+4+4+4+4
28
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
page - 5
In words
Representation
Emphasis should be placed
on the number of objects.
five
plus
two groups of four
plus
one
5+2•4+1
Expression
Can you think
of a shortcut
to evaluate this
expression?
5+4+4+1
14
Expression with addition only
Answer
In words
Representation
Emphasis should be placed
on the number of objects.
four groups of one
plus
five groups of two
4•1+5•2
Expression
1+ 1+ 1+ 1+ 2 + 2+ 2+ 2+ 2
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Can you think
of a shortcut
to evaluate this
expression?
14
Answer
page - 6
In words
Representation
Emphasis should be placed
on the number of objects.
ten groups of four
plus
nine
10 • 4 + 9
Expression
Can you think
of a shortcut
to evaluate this
expression?
4+4+4+4+4+4+4+4+4+4+9
49
Expression with addition only
Answer
In words
Representation
Emphasis should be placed
on the number of objects.
five groups of two
plus
seven groups of two
5•2+7•2
Expression
2+2+2+2+2 +
2+2+2+2+2+2+2
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Can you think
of a shortcut
to evaluate this
expression?
24
Answer
page - 7
Multiplication, Addition and Parenthesis
In words
Representation
Emphasis should be placed
on the number of objects.
two groups of
three plus one
2 ( 3 + 1)
Expression
3+1+ 3+1
8
Expression with addition only
Answer
In words
Representation
Emphasis should be placed
on the number of objects.
four groups of
two plus three
4 ( 2 + 3)
Expression
2+3 + 2+3 + 2+3 +2+3
20
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
page - 8
In words
Representation
Emphasis should be placed
on the number of objects.
three groups of
three plus two
plus
six
3 ( 3 + 2) + 6
Expression
Can you think
of a shortcut
to evaluate this
problems?
3+2 + 3+2 + 3+2 + 6
Expression with addition only
Answer
In words
Representation
21
Emphasis should be placed
on the number of objects.
six
plus
four groups of
one plus one
6 + 4 (1 + 1)
Expression
Can you think
of a shortcut
to evaluate this
expression?
6 + 1+1 + 1+1 + 1+1 +1+1
14
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
page - 9
In words
Representation
Emphasis should be placed
on the number of objects.
four groups of two
plus
two groups of
six plus two
4 • 2 + 2 ( 6 + 2)
Expression
Can you think
of a shortcut
to evaluate this
problems?
2+2+2+2 + 6+2+6+2
24
Expression with addition only
Answer
In words
Representation
seven groups of
four plus one
plus
four groups of three
7 (4 +1) + 4 • 3
Expression
4+1+4+1+4+1+4+1+
4+1+4+1+4+1 + 3+3+3+3
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Can you think
of a shortcut
to evaluate this
problems?
47
Answer
page - 10
In words
Emphasis should be placed
on the number of objects.
Representation
three groups of
two groups of four
3•2•4
Expression
Can you think
of a shortcut
to evaluate this
problems?
4+4 + 4+4 +4+4
24
Expression with addition only
Answer
Emphasis should be placed
on the number of objects.
In words
Representation
six groups of
two groups of three
6•2•3
Expression
3 + 3 + 3+ 3 + 3 + 3 +
3 + 3 + 3+ 3 + 3 + 3
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Can you think
of a shortcut
to evaluate this
problems?
36
Answer
page - 11
In words
Representation
three groups of
three groups of three
3
3
3
3
3
3
3
3
3
3•3•3
Expression
Can you think
of a shortcut
to evaluate this
problems?
3+3+3+3+3+3+3+3+3
27
Expression with addition only
Answer
In words
Representation
four plus
three groups of
two groups of five
4
53
5
53
5
53
5
4+3•2•5
Expression
Can you think
of a shortcut
to evaluate this
problems?
4 + 5+5+5+5+5+5
34
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
page - 12
In words
Representation
5
five times five
therefore
5
5
5
5
five groups of five
52
Expression
5+5+5+5+5
25
Expression with addition only
Answer
In words
Representation
three times three
3
3
therefore
3
three groups of three
32
Expression
3+3+3
9
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
page - 13
In words
Representation
four groups of four
plus
ten
4
4
4
10
4
4 2 + 10
Expression
Can you think
of a shortcut
to evaluate this
problems?
4+4+4+4 + 10
26
Expression with addition only
Answer
In words
Representation
three groups of three
plus
three groups of five
3
3
3
5
5
5
3 2+3 • 5
Expression
Can you think
of a shortcut
to evaluate this
problems?
3+3+3+5+5+5
24
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
page - 14
In words
Representation
one plus
six groups of six
plus
two groups of three
1
6
6
6
3
6
6
6
3
1+62+2•3
Expression
Can you think
of a shortcut
to evaluate this
problems?
1 + 6+6+6+6+6+6 + 3+3
43
Expression with addition only
Answer
In words
Representation
two groups of two
plus
three groups of
three plus one
2
2
3 1 3 1
3 1
2 2 + 3 ( 3 + 1)
Expression
Can you think
of a shortcut
to evaluate this
problems?
2+2 + 3+1 + 3+1 + 3+1
16
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
page - 15
In words
Representation
three
plus
six groups of six
plus
three groups of three
3
6
6
6
6
6
6
3
3
3
3+62+32
Expression
Can you think
of a shortcut
to evaluate this
problems?
3 + 6+ 6+ 6+ 6+ 6+ 6+
3+ 3 + 3
48
Expression with addition only
Answer
In words
Representation
nine groups of nine
plus
five groups
of
four plus seven
9 9 9
9 9 9
9
9
4 7
4 7 4 7
4 7
9
4 7
9 2 + 5 ( 4 + 7)
Expression
9+9+9+9+9+9+9+9+9+
4+7 + 4+7 + 4+7 + 4+7 + 4+7
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Can you think
of a shortcut
to evaluate this
problems?
136
Answer
page - 16
In words
Representation
Sabrina has five nickels and ten
dimes. How much money does
Sabrina have?
5
5
5
10 10 10 10
10 10 10 10
10 10
5
5
5 • 5 + 10 • 10
Expression
5+5+5+5+5
+10+10+10+10+
10+10+10+10+10+10
Can you think
of a shortcut
to evaluate this
problems?
125 cents or $1.25
Expression with addition only
Answer
In words
Representation
Sabrina has ten nickels, five
quarters and three pennies.
How much money does
Sabrina have?
5 5
5 5
5
5
5
5
25 25
25 25
1
5
5
25
1
10 • 5 + 5 • 25 + 3 • 1
Expression
5+5+5+5+5+5+5+5+5+5+
25+25+25+25+25+
1+1+1
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
Can you think
of a shortcut
to evaluate this
problems?
178
page - 17
1
In words
Representation
Sabrina has seven eggs in her
refrigerator. She buys three more
dozens eggs. How many eggs
does she have?
7 + 3 • 12
Expression
Can you think
of a shortcut
to evaluate this
problems?
7 + 12 + 12 + 12
43 eggs
Expression with addition only
Answer
In words
Representation
Sabrina has three boxes of crayons
with eight crayons in each box.
She buys six more boxes with ten
crayons in each box. How many
crayons does she have?
8 crayons
10
10
10
10
10
10
8 crayons
8 crayons
3 • 8 + 6 • 10
Expression
Can you think
of a shortcut
to evaluate this
problems?
8+8+8+
10 +10 +10 +10 +10 +10
Expression with addition only
© S. Mercer - Order of Operations Answer Key
Answer
64 crayons
page - 18
Subtraction
In words
Representation
take away five
from
twelve
twelve minus five
12 - 5
Expressi
on
12 – 5
7
Expression with addition
and/or subtraction only
Answer
In words
Representation
take away nine
from
ten
ten minus nine
10 - 9
Expressi
on
10 – 9
Expression with addition
and/or subtraction only
© S. Mercer - Order of Operations Answer Key
1
Answer
page - 19
In words
Representation
take away
five
from
two groups of three
2•3-5
Expression
3+3–5
1
Expression with addition and/or
subtraction only.
Answer
In words
Representation
take away
two groups of four
from
ten
10 - 2 • 4
Expression
10 – 4 – 4
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations Answer Key
2
Answer
page - 20
In words
Representation
take away
five groups of two
from
seven groups of two
7•2-5•2
Expression
2+2+2+2+2+2+2–
2–2–2–2–2
Can you think
of a shortcut
to evaluate this
problems?
4
Expression with addition and/or
subtraction only.
Answer
In words
Representation
take away
four groups of four
from
six groups of three
6•3-4•4
Expression
3+3+3+3+3+3–
4–4–4–4
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations Answer Key
Can you think
of a shortcut
to evaluate this
problems?
2
Answer
page - 21
In words
Representation
two groups
of
seven minus three
2 ( 7 - 3)
Expression
7–3+7–3
8
Expression with addition and/or
subtraction only.
Answer
In words
Representation
four groups
of
nine minus five
4 ( 9 - 5)
Expression
9–5 + 9–5 + 9–5 + 9–5
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations Answer Key
16
Answer
page - 22
In words
Representation
Sabrina has six dimes and ten
quarters. She spends fifty cents.
How much money does Sabrina
have left?
10 10
10 10
25
25
25 25
25
25
25 25
10 10
25
25
6 • 10 + 10 • 25 – 50
Expression
10+10+10+10+10+10
25+25+25+25+25+25+2
5+25+25+25 – 50
Expression with addition and/or
subtraction only.
260 cents or $2.60
Answer
In words
Representation
Sabrina has ten nickels, one quarter
and ten dimes. She spends 75
cents. How much money does
Sabrina have left?
5
5
5
5
5
5
5
5
5
5
10 10 10
25
10
10 10
10
10 10
10
10•5 + 1•25 + 10•10 –75
Expression
5+5+5+5+5+5+5+5+5+5+
25+10+10+10+10+10+10
+10+10+10+10 – 75
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations Answer Key
100 cents or $1.00
Answer
page - 23
In words
Representation
Sabrina has six boxes of crayons,
with eight crayons in each box. She
uses ten crayons. How many
crayons does she have left?
6 • 8 – 10
Expression
8+8+8+8+8+8
– 10
38 crayons
Expression with addition and/or
subtraction only.
Answer
In words
Representation
Sabrina bought two dozen eggs and
uses eight to do an omelette. How
many eggs does she have left?
2 • 12 – 8
Expression
16 eggs
12 + 12 – 8
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations Answer Key
Answer
page - 24
In words
Representation
Sabrina has 10 cans of soda.
She buys seven 6-packs of soda.
She drinks two sodas. How many
sodas does she have left?
6 cans
6 cans
6 cans
6 cans
6 cans
6 cans
6 cans
10 + 7 • 6 – 2
Expression
10 + 6 + 6 + 6 + 6 +
6+6+6–2
Expression with addition and/or
subtraction only.
In words
50 cans
Answer
Representation
Sabrina has five dozen pens. She
gives away two dozen to her
friends. How many pens does she
have left?
12
pens
12
pens
12
pens
12
pens
12
pens
5 • 12 – 2 • 12
Expression
12 + 12 +12 +12 +12
– 12 – 12
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations Answer Key
36 pens
Answer
page - 25
Order
of
Operations
Name: ____________________
Period: ________
Date: ______________________
Multiplication
In words
Representation
3 • 7
Expression
Expression with addition only
Answer
In words
Representation
4 • 5
Expression
Expression with addition only
© S. Mercer - Order of Operations
Answer
page - 1
In words
Representation
3 • 3
Expression
Expression with addition only
Answer
In words
Representation
1 • 9
Expression
Expression with addition only
© S. Mercer - Order of Operations
Answer
page - 2
Multiplication and Addition
In words
Representation
5+2 • 3
Expression
Expression with addition only
Answer
In words
Representation
9+ 3 • 4
Expression
Expression with addition only
© S. Mercer - Order of Operations
Answer
page - 3
In words
Representation
7•4+2
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
10 + 6 • 2
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 4
In words
Representation
3+2•8+2
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
2•6+4•4
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 5
In words
Representation
5+2•4+1
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
4•1+5•2
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 6
In words
Representation
10 • 4 + 9
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
5•2+7•2
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 7
Multiplication, Addition and Parenthesis
In words
Representation
2 ( 3 + 1)
Expression
Expression with addition only
Answer
In words
Representation
4 ( 2 + 3)
Expression
Expression with addition only
© S. Mercer - Order of Operations
Answer
page - 8
In words
Representation
3 ( 3 + 2) + 6
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
6 + 4 (1 + 1)
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 9
In words
Representation
4 • 2 + 2 ( 6 + 2)
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
7 (4 + 1) + 4 • 3
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 10
In words
Representation
3•2•4
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
6•2•3
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 11
In words
Representation
3•3•3
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
4+3•2•5
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 12
In words
Representation
52
Expression
Expression with addition only
Answer
In words
Representation
32
Expression
Expression with addition only
© S. Mercer - Order of Operations
Answer
page - 13
In words
Representation
4 2 + 10
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
3 2+3 • 5
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 14
In words
Representation
1+62+2•3
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
2 2 + 3 ( 3 + 1)
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 15
In words
Representation
3+62+32
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
9 2 + 5 ( 4 + 7)
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 16
In words
Representation
Sabrina has five nickels and ten
dimes. How much money does
Sabrina have?
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
Sabrina has ten nickels, five
quarters and three pennies.
How much money does
Sabrina have?
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 17
In words
Representation
Sabrina has seven eggs in her
refrigerator. She buys three more
dozens eggs. How many eggs
does she have?
Expression
Expression with addition only
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
Sabrina has three boxes of crayons
with eight crayons in each box.
She buys six more boxes with ten
crayons in each box. How many
crayons does she have?
Expression
Expression with addition only
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 18
Subtraction
In words
Representation
12 - 5
Expression
Expression with addition and/or
subtraction only
Answer
In words
Representation
10 - 9
Expression
Expression with addition
and/or subtraction only
© S. Mercer - Order of Operations
Answer
page - 19
In words
Representation
2•3-5
Expression
Expression with addition and/or
subtraction only.
Answer
In words
Representation
10 - 2 • 4
Expression
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations
Answer
page - 20
In words
Representation
7•2-5•2
Expression
Expression with addition and/or
subtraction only.
Answer
In words
Representation
Can you think
of a shortcut
to evaluate this
expression?
6•3-4•4
Expression
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations
Can you think
of a shortcut
to evaluate this
expression?
Answer
page - 21
In words
Representation
2 ( 7 - 3)
Expression
Expression with addition and/or
subtraction only.
Answer
In words
Representation
4 ( 9 - 5)
Expression
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations
Answer
page - 22
In words
Representation
Sabrina has six dimes and ten
quarters. She spends fifty cents.
How much money does Sabrina
have left?
Expression
Expression with addition and/or
subtraction only.
Answer
In words
Representation
Sabrina has ten nickels, one quarter
and ten dimes. She spends 75
cents. How much money does
Sabrina have left?
Expression
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations
Answer
page - 23
In words
Representation
Sabrina has six boxes of crayons,
with eight crayons in each box.
She uses ten crayons. How many
crayons does she have left?
Expression
Expression with addition and/or
subtraction only.
In words
Answer
Representation
Sabrina bought two dozen eggs
and uses eight to do an omelette.
How many eggs does she have
left?
Expression
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations
Answer
page - 24
In words
Representation
Sabrina has 10 cans of soda.
She buys seven 6-packs of soda.
She drinks two sodas. How many
sodas does she have left?
Expression
Expression with addition and/or
subtraction only.
In words
Answer
Representation
Sabrina has five dozen pens. She
gives away two dozen to her
friends. How many pens does she
have left?
Expression
Expression with addition and/or
subtraction only.
© S. Mercer - Order of Operations
Answer
page - 25