Simplifying Numerical Expressions – Grade Five
Ohio Standards
Connection
Number, Number Sense
and Operations
Benchmark E
Use order of operations,
including use of
parenthesis and exponents
to solve multi-step
problems, and verify and
interpret the results.
Indicator 9
Use order of operations,
including use of
parentheses, to simplify
numerical expressions.
Related Benchmark
F. Apply number system
properties when
performing
computations.
Mathematical Processes
Benchmarks
F. Use inductive thinking
to generalize a pattern
of observations for
particular cases, make
conjectures, and
provide supporting
arguments for
conjectures.
H. Use representations to
organize and
communicate
mathematical thinking
and problem solutions.
Lesson Summary:
In this lesson, students learn about the order of operations and
how to use the order of operations to simplify expressions.
First, students learn the parenthesis signifies which operation is
executed first when simplifying expressions. Then, students
analyze simplified expressions, write the steps needed to
simplify the expression and then generalize the order for which
operations are executed. In journals, mathematical terminology
and summaries of understanding are recorded.
Estimated Duration: Two hours
Commentary:
Students learning about the order of operations are typically
presented the pneumonic, “Please Excuse My Dear Aunt Sally.”
However, this technique has caused misconceptions and
confusion for many students. In this lesson, students discover
the order through analysis of simplified expressions. They can
make and test conjectures that will engage them in the
understanding and use of the order of operations.
Pre-Assessment:
Assess strategies for solving multiple step problems and writing
equations to represent the situation.
 Pose the following situation to the students.
The teacher ordered gifts for the 20 students in the
classroom. The mini notebooks cost $1.50 and the dazzling
pens cost $2.50.
 Direct students to determine the cost for the gifts and write
an expression to represent the situation.
 Select students to share their determined cost and
expressions. Ask students questions to explain the methods
used.
 List the expressions created by students on the board.
Scoring Guidelines:
Assess students’ problem solving skills and understanding of
the operations. The process should include an acceptable way to
determine the cost for all of the gifts. Note students who
multiply each price by 20 and then add, and those that multiply
the sum of the items by 20.
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Simplifying Numerical Expressions – Grade Five
Ohio Standards
Connection
Post-Assessment:
Have students simplify a variety of expressions and write the
steps for simplifying.
 Distribute Simplifying Numerical Expressions PostAssessment, Attachment A.
Scoring Guidelines:
Use the rubric to determine the level of students understanding
and progress. Attachment B contains the simplification and steps.
Score
Description
Meeting
Simplifies all expressions accurately
Expectations Provides the appropriate steps for simplifying
Progressing Simplifies expressions with minor flaws
Adequately Provides appropriate steps for simplifying
Intervention Provides evidence of partial understanding
Needed
Executes the same minor errors throughout all
simplifications.
Instruction Uses a random order to simplify each expression
required
or simplifies from left to right regardless of
operation.
Instructional Procedures:
Part One
1. Direct students to look at the equations on the board.
20  $1.50 + $2.50 = $80
$1.50 + $2.50  20 = $80
a. Describe the problem situation from the pre-assessment
and relate it to the first equation, “Twenty students
multiplied by the cost of the mini notebook and the
dazzling pen.”
b. Describe the situation as it relates to the second equation,
“The cost of the mini notebook and the dazzling pen
multiplied by 20.”
c. Have students check the equations for accuracy. As
written, in arithmetic thinking, the answer to the first
equation is $32.50.
d. Explain to the students that, in mathematics, parentheses
are used to signify an order for which to simplify the
expression.
e. Rewrite the first equation. Tell students that in this
equation the parenthesis are used to signify the first step in
solving the expression.
20 ($1.50 + $2.50)
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Simplifying Numerical Expressions – Grade Five
2.
3.
4.
5.
6.
7.
f. Ask students how they would change the second equation to signify which operation to
complete first.
Pose the following situation:
The seven cheerleaders found sweaters priced at $15 on sale. The cost of each sweater was
then reduced by $3. Determine the cost of the seven sweaters and write an expression using
parenthesis to represent the situation.
Observe students as they complete the task and assist as needed. Note students who begin to
use the parenthesis appropriately and those who multiply seven and the difference of the
regular price and the three dollar reduction ($12).
Select students to share their responses and equations. Write the expression on the board and
reinforce the significance of the parenthesis.
7 ($15 - $3) =
Present number expressions for students to simplify for practice.
 5 + (3  7) (26)
 21 – (12  6) (19)
 56 – (7  6) (14)
Introduce vocabulary to students.
 expression (any combination of numbers and symbols, excluding the equality or
inequality symbols)
 simplify (to perform all operations within an expression, like solving an equation)
a. Have students write the terms in their mathematics journal. Have students discuss each
term.
b. Ask questions such as,
 What is it?
 What is it like? or not like?
 Can you give an example?
Have students write statements, in a mathematics journal, about the significance of
parenthesis in simplifying expressions. Provide exercises to practice for homework.
Part Two
8. Review the use of parenthesis in simplifying expressions.
9. Present Finding the Order of Operations, Attachment B, to students and place them into
groups of four. The groups analyze several expressions and determine the steps used to
simplify the expression. Students write the steps for each expression in the boxes. See
Attachment C, Finding the Order of Operations Answer Key. Attachment B contains two
pages of expressions. Page one contains expressions using four to five operations, while page
two offers less complex expressions. Choose the expressions which best suit the needs of the
students.
10. Review the steps used to simplify each expression with the students and allow them to make
corrections as needed. Ask students questions about the steps and the order in which the
operations seem to appear.
 If applicable, which always occurs first? (Operations in parenthesis).
 Do you notice any order to the operations as they are listed in the steps? (They always
appear after the parenthesis, where applicable. It appears that division and multiplication
are used before addition and subtraction.)
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Simplifying Numerical Expressions – Grade Five

Do you notice any instances in which addition and subtraction occur before
multiplication and division? (Yes, but only when addition and subtraction occur within
parenthesis.)
 Does it appear that there is an order for division and multiplication? (No, they are
interchangeable. Yes, you complete them from left to right in the expression.)
 Does it appear that there is an order to addition and subtraction? (No, they are
interchangeable. Yes, you complete them from left to right in the expression.)
11. Introduce vocabulary to students.
 order of operations
a. Have students write the terms in their mathematics journal. Have students discuss each
term.
b. Ask questions such as,
 What is it?
 What is it like? or not like?
 Can you give an example?
12. Have the groups write statements, in a mathematics journal, about the order of operations and
how they are used to simplify expressions. Include a graphic organizer or visual.
13. Assign expressions for students to practice simplifying using the order of operations.
Differentiated Instructional Support:
Instruction differentiated according to learner needs, to help all learners either meet the intent of
the specified indicator(s) or, if the indicator is already met, to advance beyond the specified
indicator(s).
 Use calculators, number lines, and factor charts as tools for students who are learning basic
facts.
 Present expressions with fractions, decimals and percents.
 Have students write expressions to represent different situations they may face in daily life.
Extensions:
 Have the students create expressions for stories, and then simplify using order of operations.
 Using 2, 4, 6, 8, 9 create an expression using all four operations that will obtain the largest
possible value with the given set of numbers. Create an expression that will obtain the least
possible value with the given set of numbers.
 Write and simplify expressions that contain a variable.
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Simplifying Numerical Expressions – Grade Five
Home Connections:
Discuss with parent and ask what has happens to a recipe when the measurements of an
ingredient have been changed, such as only making half of a recipe or when you double a recipe.
Materials and Resources:
The inclusion of a specific resource in any lesson formulated by the Ohio Department of
Education should not be interpreted as an endorsement of that particular resource, or any of its
contents, by the Ohio Department of Education. The Ohio Department of Education does not
endorse any particular resource. The Web addresses listed are for a given site’s main page,
therefore, it may be necessary to search within that site to find the specific information required
for a given lesson. Please note that information published on the Internet changes over time,
therefore the links provided may no longer contain the specific information related to a given
lesson. Teachers are advised to preview all sites before using them with students.
For the students: Calculators, journals
Vocabulary:
 expression
 order of operations
 simplify
Technology Connections:
 Identify the difference between Algebraic and Arithmetic calculators. Algebraic calculators
follow the order of operations, where Arithmetic calculators do not follow the order of
operations.
 Have students use spreadsheets when looking at different sequences.
 Have students use mathematics software that would enhance the lesson on use of order of
operations
Research Connections:
"BSCS Science: An Inquiry Approach." BSCS Biological Sciences Curriculum Study. 23 Dec.
2003 <http://63.225.114.218/bscsdotorg/curriculum/InquiryFAQs.htm>
Marzano, Robert J., Jane E. Pollock and Debra Pickering. Classroom Instruction that Works:
Research-Based Strategies for Increasing Student Achievement, Alexandria, VA: Association for
Supervision and Curriculum Development, 2001
Attachments:
Attachment A, Simplifying Numerical Expressions Post-Assessment
Attachment B, Simplifying Numerical Expressions Post-Assessment Answer Key
Attachment C, Finding the Order of Operations Cards
Attachment D, Finding the Order of Operations Cards Answer Key
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Simplifying Numerical Expressions – Grade Five
Attachment A
Simplifying Numerical Expressions Post-Assessment
Name_______________________________
Date__________________________
Directions: Simplify the expressions and write the steps used in the process.
1. Simplify the expression:
(4 + 7)  3
Write the steps used to simplify.
2. Simplify the expression:
32  8 + 5  8
Write the steps used to simplify.
3. Simplify the expression:
40 + (6  5)  3
Write the steps used to simplify.
4. Simplify the expression:
36  6  3  (9  4)
Write the steps used to simplify.
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Simplifying Numerical Expressions – Grade Five
Attachment B
Simplifying Numerical Expressions Post-Assessment Answer Key
1. Simplify the expression:
(4 + 7)  3
33
Write the steps used to simplify.
1. Add 4  7
2. Multiply 11 and 3
2. Simplify the expression:
32  8 + 5  8
44
Write the steps used to simplify.
1. Divide, 32  8
2. Multiply 5 and 8
3. Add 4 and 40
3. Simplify the expression:
40 + (6  5)  3 50
Write the steps used to simplify.
1. Multiply 6 and 5
2. Divide 30 by 3
3. Add 40 and 10
4. Simplify the expression:
36  6  3  (9  4)
23
Write the steps used to simplify.
1. Subtract 9  
2. Divide 36 by 6
3. Multiply 6 and 3
4. Add 18 and 5
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Simplifying Numerical Expressions – Grade Five
Attachment C
Finding the Order of Operations Cards
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Simplifying Numerical Expressions – Grade Five
Attachment C (continued)
Finding the Order of Operations Cards
9
Simplifying Numerical Expressions – Grade Five
Attachment D
Finding the Order of Operations Answer Key
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Simplifying Numerical Expressions – Grade Five
Attachment D (continued)
Finding the Order of Operations Answer Key
11