Lesson One – Parentheses in Number Sentences
I.
Lesson Rationale
Learning Goals and Focus:
This lesson is intended as a review of parentheses for students and will serve as an
introduction to the order of operations. In this lesson students will use parentheses in
number sentences involving more than one operation. They will learn to identify and
write sentences that model number stories. They will solve problems involving
parentheses and nested parentheses. They will also insert parentheses in order to make
true number sentences.
Illinois Common Core Standards:
CCSS.Math.Content.5.OA.A.1 Use parentheses, brackets, or braces in numerical
expressions, and evaluate expressions with these symbols.
CCSS.Math.Content.5.OA.A.2 Write simple expressions that record calculations with
numbers, and interpret numerical expressions without evaluating them. For
example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7).
Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without
having to calculate the indicated sum or product.
Academic Language Target:
Students will be able to identify and use the following vocabulary: expression,
ambiguous, nested parentheses, parentheses, and number sentences.
II.
Instructional Strategies and Learning Tasks
Activating background knowledge (5 minutes):
Since mathematics is the first class of the day and is scheduled for 60 minutes,
students arrive to school and get ready to begin class. As part of our classroom routine,
a problem of the day is posted on the ELMO projector and students know they are to
begin answering the question in their mathematic notebooks. As instructed by the
school and my cooperating teacher our curriculum follows the Everyday Mathematics
text. First, I will pose the question: ‘What do parentheses mean in number sentences?’
From student responses, I hope a student will be able to tell say operations inside
parentheses are done first.
Explicit Instruction (15 minutes):
I have learned from my cooperating teacher the importance of writing notes as a
way for students to have a point of reference and something to refer back to when
working independently on a problem. As a class I will write the following for students
to copy into their mathematical notebooks: ‘Without mathematical punctuation of
parentheses, number expressions can take on different values depending on the order
in which the operations are performed. Without parentheses, the expression is said to
be ambiguous because it has more than one possible meaning.’ After copying this into
their mathematical notebooks, I would want to discuss the term ambiguous. I know the
majority of my students will not know what this term means and I want to discuss
possible synonyms as a class that will be clear and easy for the students to remember.
The synonym I have in mind for ambiguous is unclear. By using a vocabulary strategy
like finding an antonym or synonym for unfamiliar words it makes it easier for students
to remember.
Students would continue to copy into their notebooks the following: ‘An expression
is a group of mathematical symbols (numbers, operation signs, variables, and grouping
symbols) that represents the number.’ I would stop between copying this on the
overhead and ask students if they knew what I meant when I said mathematical
symbols to see if they could come up with their own examples instead of just listing
them directly. It is important for students to make their own connections to the
material through questioning techniques.
Structured Practice (15 minutes):
After discussing possible answers with the class, I will pose the following problem:
6 * 4 – 2 / 2 = n. I would ask students to work in the pairs (the student sitting closest to
them) to find answers to the problem listed above. I would ask the following question:
‘What different ways could we solve this problem?’ Students will have approximately
five minutes to think of possible answers.
The following are answers I am expecting the students to find: ((6 * 4) – 2) / 2 = 11,
6 * (4-2) / 2 = 6, (6 * 4) – (2 * 2) = 23, and 6 * (4 – (2/2)) = 18. I will circulate around the
classroom to assist students with their answers. There will also be a special education
teacher in the classroom working with a group of fours students who do not have an
Individualized Education Plan but require extra assistance during mathematics.
After we are finished discussing all possible answers, we will relate the problem
back to our notes to discuss the reasons why without the use of parentheses there can
be multiple ways to solve the problem making it ambiguous. We will also add to our
previous notes stating ‘When two or more sets of parentheses are used in the same
expression, the operation inside the inner parentheses is done first. Parentheses inside
inner parentheses are referred to as nested parentheses.’
Guided Practice (15 minutes):
Students will work in small groups in their Everyday Mathematics Journals Volume
2 on pages 219 – 220. They will match number stories to appropriate expressions. As a
class we will take up answers and clarify any misunderstandings or confusions the
students may have.
Independent Practice (10 minutes):
Students will be given a photocopy of Everyday Mathematics Math Masters
workbook page 198, which focuses on reviewing parentheses. Students will also
complete Lesson 7.4 in their Everyday Mathematics Study Links Workbook. Students
will be given some time to begin their independent practice and if not completed in
class it will be taken home as homework.
III.
Assessment
Formative and Summative Assessment:
As students are working in pairs and small groups, I will be walking around the
classroom and checking in with each student to see how and what progress is being
made on the assigned task. Doing this allows me to gauge what areas need further
clarification as a whole group, if I can help the student as I’m walking around, or if the
student needs extra help during Tiered Support. Tiered Support occurs during the last
thirty minutes of the school day that allows the classroom teacher to provide any
additional support to students who may require it.
I will also be giving students the following exit slip: ‘Explain how you used
parentheses in Problem 6 on journal page 220 to write the expression for the total
number of undamaged cans.’ I will be looking to see if students can refer to the use of
nested parentheses to identify the total number of undamaged cans.
The following day I will also be using the problem of the day to review ambiguous
mathematical expressions. I will be asking the following question of students: ‘Robin
asked her friends to solve 4 + 5 * 8 =? What problems might arise from her friend’s
answers?’ I will collect the problem of the day and I will also collect the previous nights
homework to review.
IV.
Instructional Materials
-
Projector
-
Mathematical notebook to write notes and questions to be shown on the projector.
-
Everyday Mathematics Journal Volume 2, pages 219 – 220.
-
Everyday Mathematics Study Link Workbook, Lesson 7.4.
-
Photocopy of Everyday Mathematics Math Masters (Teacher’s Resource), page 198.
-
Post-it notes for exit slip.
Lesson Two – Order of Operations
I.
Lesson Rationale
Learning Goals and Focus:
During this lesson, students will be introduced to the rules that govern the order in
which operations are performed in an expression. Students will be able to evaluate
numerical expressions using the order of operations. They will also understand the
precedence of multiplication and division over addition and subtraction.
Illinois Common Core Standards:
CCSS.Math.Content.5.OA.A.1 Use parentheses, brackets, or braces in numerical
expressions, and evaluate expressions with these symbols.
Academic Language Target:
Students will become familiar with the term order of operations. They will be able
to explain the process when solving a numerical expression by using the order of
operations.
II.
Instructional Strategies and Learning Tasks
Activating background knowledge (15 minutes):
Mathematics is held during the first 60 minutes of the day. Students will get settled
into the classroom and begin to work on the problem of the day. As discussed in the
assessment section of the first lesson, the problem of the day will focus on the key
concepts surrounding ambiguous mathematical expressions. This will serve as a review
and lead the class into their next lesson involving the order of operations. The problem
will be projected on the overhead and it will state: ‘Robin asked her friends to solve 4 +
5 * 8 =? What problems might arise from her friend’s answers?’ I’m anticipating
students will come up with the following answers: some of her friends solved (4+5) * 8
and got 72; the other friends solved 4 + (5 * 8) and got 44; the expression caused
confusion because it is ambiguous.
After discussing all possible answers, I will ask students: ‘How do parentheses help
clarify ambiguous expressions?’ At this point, I would expect students to be able to tell
me that the operations inside parentheses are done first, and that the order for
computation in an expression can be shown with parentheses.
Explicit Instruction (10 minutes):
In our mathematic notebooks we will begin to copy the rules for the order of
operations. I will write the following: ‘1. Parentheses 2. Exponents 3. Multiplication and
Division 4. Addition and Subtraction.’ I would also stop and ask students if they already
knew what some of the rules are from previous grades where they might have
encountered the order of operations. I would then continue to write the following: ‘The
order of operations eliminates ambiguity in number expressions by providing the steps
used to evaluate them. 1. Do any operations inside parentheses first. If there are nested
parentheses, start with the innermost set of parentheses. To determine the order of
operations inside parentheses, use steps 2-4. 2. Calculate exponents in order from left
to right. 3. Multiply and divide. Neither multiplication nor division has priority over the
other simply work left to right. 4. Add and subtract. Neither addition nor subtraction
has priority over the other; simply work from left to right.’ I would also ask students
along the way to see if they knew what each step involved. I believe many students have
not experienced numerical expressions that involve nested parentheses or exponents. I
would then ask if anyone knew the mnemonic device and write: ‘Mnemonic device:
Please Excuse My Dear Aunt Sally (PEMDAS).’
Guided Practice (30 minutes):
In small-differentiated groups, students would be given a problem with a varying
degree of difficulty. For example, my above-level learners might be given an expression
with nested parentheses and exponents, while my below-level learners might be given
an expression with only multiplication, division, subtraction, and addition. In their
groups students would not only work to solve the problem but explain how they
decided to solve the problem by writing a step-by-step explanation describing how they
used the order of operations. The questions I will use are as follows: 4 + 5 * 6 =?, 3 * 10
/ 5 + 18/3 = ?, (4+5) * (2 + 3) – (10* 2) =?, ሺ5 + 5ሻଶ = ?, ሺ10ଶ + (3 * 8)) -14 =?, each
question increasing in difficulty. There are key steps I want to make sure the students
are incorporating into their answers like making sure to first evaluate the problem, to
work from left to right on exponents, multiplication and division, as well as addition
and subtraction, and if there were nested parentheses to begin with innermost pair
first.
As students worked in small groups, I would circulate around the classroom
checking in with each group and ask guiding questions to help them thoroughly explain
what steps they took to solve the problem. After about 20 minutes and students have
completed the problem and their explanations a representative from each group would
come to the projector and present their problem to the class. I always try to incorporate
students explaining their reasoning to the class as much as possible. It empowers
students to take responsibility for their learning and help others in the class who may
need extra guidance. When students are presenting, we go through each problem, make
sure all steps and language are correct, and answer any remaining questions.
Independent Practice (5 minutes):
Students will complete Lesson 7.5 in their Everyday Mathematics Study Links
Workbook. Students will also complete page 223 in the Everyday Mathematics Journal
Volume 2. Students will be given some time to begin their independent practice and if
not completed in class it will be taken home as homework.
III.
Assessment
Formative and Summative Assessment:
As students are working in their small groups, I will be walking around the
classroom and checking in with each group to see how and what progress is being made
on the assigned task. Also, I will be collecting the group’s worksheets to see if any
responses need further clarification in class the next day.
During group work it can be difficult to distinguish if a student is struggling with an
idea or concept. For the following lesson during the problem of the day I will ask
students to solve the following: ‘Solve the following problem. Explain how you found
your answer using the order of operations. 12 x 2 + 8 / 2 =?’ I will also be collecting and
reviewing the student’s homework from today’s lesson.
IV.
-
Instructional Materials
Projector
-
A differentiated problem for each group to work on.
-
Mathematical notebook to write notes and questions to be shown on the projector.
-
Everyday Mathematics Journal Volume 2, page 223.
-
Everyday Mathematics Study Link Workbook, Lesson 7.5.
-
Loose-leaf for problem of the day.
Lesson Three – Order of Operations Number Stories
I.
Lesson Rationale
Learning Goals and Focus:
During this lesson, students will learn how to write and solve their own number
stories using the order of operations.
Illinois Common Core Standards:
CCSS.Math.Content.5.OA.A.1 Use parentheses, brackets, or braces in numerical
expressions, and evaluate expressions with these symbols.
Academic Language Target:
Students will become familiar with the term order of operations in terms of writing
and solving their own number story.
II.
Instructional Strategies and Learning Tasks
Activating background knowledge (15 minutes):
Students will get settled into the classroom and begin to work on the problem of the
day. As discussed in the assessment section of the second lesson, the problem of the day
will focus on how students solve a numerical expression and explain how they solved
the problem using the order of operations. This will serve as a review and lead the class
into their assignment of creating their own number stories in small groups. The
problem will be projected on the overhead and it will state: ‘Solve the following
problem. Explain how you found your answer using the order of operations. 12 x 2 + 8 /
2 =?’ While students are working on the problem of the day, I will circulate around the
room to see how students are making progress on their questions.
I will ask for volunteers to come to the projector and explain what steps they took to
solve their answer. For this, portion I ask students to put away their pencils and take
out their pens since I will be collecting the problem of the day at the end of the class. I
ask students to take out pens so they can make corrections to their papers but also so I
can see what their thinking was like when they initially solved the problem. It allows
me to see what misunderstanding may have occurred and where clarifications need to
be made to help the student succeed with the mathematical concept.
Explicit Instruction (10 minutes):
After the answer to the problem of the day has been discussed, I will break the
students into mixed groups making sure there are a variety of learners in each group.
Once the students are in their groups, I will explain they will be creating their own
number stories like we have encountered in the Everyday Mathematic Journals Volume
2 for the last two lessons. I will encourage students to use the number stories in their
journals as a guide but to be creative. I will tell them they can create their own stories
but it must include a problem that involves the order of operations. Next we will discuss
possible strategies to help them create their own stories. For example, I would tell them
that sometimes it is easier to create the numerical expression and then create a story
afterwards to fit the expression.
Independent Practice (35 minutes):
Students will take the knowledge they have gained from the past two lessons and
apply it to this activity. They will work independently in their groups and create their
own number stories. I will circulate the classroom and ask guiding questions to any
group who might be struggling with the idea or concept. As groups begin to finish their
number stories, I will have them exchange with another group and have the groups try
and solve the number stories. If there is not enough time to have an exchange of
number stories, some will be selected for the next problem of the day.
III.
Assessment
Formative and Summative Assessment:
As students are working in their small groups, I will be walking around the
classroom and checking in with each group to see how and what progress is being made
on the assigned task. Also, I will be collecting the groups’ worksheets to check their
progress and see if they were able to create a number story using the order of
operations.
A quiz will be given the following day as an assessment to see how students are
grasping the material from lessons one through three.
IV.
-
Instructional Materials
Loose-leaf for problem of the day and activity
Week: Mar 18 - 22
MONDAY
Essential
Questions:
Vocabulary:
EM Lesson: Review/Quiz
How do I write
numbers in standard
and exponential
notation?
How do I explore
place value using
powers of 10?
How do I write and
translate numbers in
and between standard
and exponential
notation?
How do I explore the
place value of
numbers written as
powers of 10?
How do I translate
numbers from
scientific notation to
standard and numberand-word notation?
-
Standard notation
Exponential notation
Base
Exponent
Factor
Power of a number
Number-and-word
notation
TUESDAY
-
-
-
-
EM Lesson: 7.4
How do I identify and
write sentences that
model number
stories?
How do I solve
problem involving
parentheses and
nested parentheses?
How do I insert
parentheses in order
to make true number
sentences?
Expression
Ambiguous
Nested Parentheses
Parentheses
Number sentences
WEDNESDAY
-
-
-
EM Lesson: 7.5
How do I evaluate
numerical expressions
using order of
operations?
How do I use the
precedence of
multiplication and
division over addition
and subtraction?
Order of operations
THURSDAY
-
-
EM Lesson: 7.6
How do I write and
solve a number story
using the order of
operations?
Expression
Ambiguous
Nested Parentheses
Parentheses
Number sentences
Order of operations
FRIDAY
EM Lesson: Quiz
How do I evaluate
numerical
expressions using
order of
operations?
How do I write
numbers in
standard and
exponential
notation?
How do I explore
place value using
powers of 10?
How do I write
and translate
numbers in and
between standard
and exponential
notation?
How do I explore
the place value of
numbers written
as powers of 10?
How do I translate
numbers from
scientific notation
to standard and
number-and-word
notation?
-
Standard notation
Exponential
notation
Base
Exponent
Factor
Power of a
number
Week: Mar 18 - 22
-
Powers of 10
Negative Exponents
Powers of 0.1
Scientific notation
Expanded notation
-
Number-and-word
notation
Powers of 10
Negative
Exponents
Powers of 0.1
Scientific notation
Expanded
notation
Expression
Ambiguous
Nested
Parentheses
Parentheses
Number sentences
Order of
operations
Week: Mar 18 - 22
CCSS.Math.Content.5.NBT.A.
1 Recognize that in a multidigit number, a digit in one
place represents 10 times as
much as it represents in the
place to its right and 1/10 of
what it represents in the place
to its left.
Core
Standard(s):
CCSS.Math.Content.5.NBT.A.
2 Explain patterns in the
number of zeros of the product
when multiplying a number by
powers of 10, and explain
patterns in the placement of the
decimal point when a decimal
is multiplied or divided by a
power of 10. Use wholenumber exponents to denote
powers of 10.
CCSS.Math.Content.5.OA.A.1
Use parentheses, brackets, or
braces in numerical
expressions, and evaluate
expressions with these
symbols.
CCSS.Math.Content.5.OA.A.2
Write simple expressions that
record calculations with
numbers, and interpret
numerical expressions without
evaluating them. For example,
express the calculation “add 8
and 7, then multiply by 2” as 2
× (8 + 7). Recognize that 3 ×
(18932 + 921) is three times as
large as 18932 + 921, without
having to calculate the
indicated sum or product.
CCSS.Math.Content.5.OA.A.1
Use parentheses, brackets, or
braces in numerical
expressions, and evaluate
expressions with these symbols.
CCSS.Math.Content.5.OA.A.1
Use parentheses, brackets, or
braces in numerical
expressions, and evaluate
expressions with these
symbols.
CCSS.Math.Content.5.OA.
A.1 Use parentheses,
brackets, or braces in
numerical expressions, and
evaluate expressions with
these symbols.
CCSS.Math.Content.5.OA.
A.2 Write simple
expressions that record
calculations with numbers,
and interpret numerical
expressions without
evaluating them. For
example, express the
calculation “add 8 and 7,
then multiply by 2” as 2 ×
(8 + 7). Recognize that 3 ×
(18932 + 921) is three
times as large as 18932 +
921, without having to
calculate the indicated sum
or product.
CCSS.Math.Content.5.NB
T.A.2 Explain patterns in
the number of zeros of the
product when multiplying
a number by powers of 10,
and explain patterns in the
placement of the decimal
point when a decimal is
multiplied or divided by a
power of 10. Use wholenumber exponents to
denote powers of 10.
Week: Mar 18 - 22
Review Lesson:
The students will be broken
into six groups. Each group
will be given a different
problem to complete on paper.
The students must solve the
problems with their group.
Each group will come to the
Elmo projector and explain
their problem to the rest of the
class. Each member of the
group will need to speak at
least once to help with the
explanation of how they solved
their problem.
•
Instruction:
Problems will be
differentiated for each
group
Types of Problems:
-
Exponential Notation
Powers of 10
Number-and-word
notation
Negative powers of
10
Expanded Notation
Scientific Notation
Ask: What do parentheses
mean in number sentences?
Operations inside parentheses
are done first.
Problem: 6*4-2/2 =?
Without mathematical
punctuation of parentheses,
number expressions can take
on different values depending
on the order in which the
operations are performed.
Without parentheses, the
expression is said to be
ambiguous because it has more
than one possible meaning.
An expression is a group of
mathematical symbols
(numbers, operation signs,
variables, and grouping
symbols) that represents a
number.
Example: The number
sentence 6*4-2/2 = n includes
the expression 6*4-2/2, the
variable n, and the equal
symbol.
What different ways could we
solve this problem?
POD: Robin asked her friends
to solve 4 +5*8=? What
problems might arise from her
friend’s answers?
-
Some of her friends
solves (4+5)*8 and got
72.
- The other friends
solved 4 +(5*8) and
got 44.
- The expression caused
confusion because it is
ambiguous.
Ask: How do parentheses help
clarify ambiguous expressions?
Because operations inside
parentheses are done first, the
order for computation in an
expression can be shown with
parentheses.
Rules for Order of Operations:
1.
2.
3.
Parentheses
Exponents
Multiplication and
Division
4. Addition and
Subtraction
The order of operations
eliminates ambiguity in number
expressions by providing the
steps used to evaluate them.
1.
((6*4)-2)/2 = 11
6*(4-2)/2 = 6
(6*4)-(2/2)=23
6*(4-(2/2))=18
Do any operations
inside Parentheses
first. If there are nested
parentheses, start with
the innermost set of
parentheses. To
POD:
-
Solve the following problem.
Explain how you found your
answer using the order of
operations. 12 x 2 + 8 / 2 = ?
-
Each group will create
a number story using
the order of
operations. Afterwards
groups will change
problems and try to
solve another group’s
problem.
Quiz on
exponents and
order of
operations.
Week: Mar 18 - 22
Note: When two or more sets
of parentheses are used in the
same expression, the operation
inside the inner parentheses is
done first parentheses inside
parentheses are referred to as
nested parentheses.
Matching number stories to
appropriate expressions MJ pg
219-220 in small groups.
Discussion of answers.
2.
3.
4.
determine the order of
operations inside
parentheses, use Steps
2-4.
Calculate Exponents in
order from left to right.
Multiple and Divide.
Neither multiplication
nor division has
priority over the other
simply work left to
right.
Add and Subtract.
Neither addition nor
subtraction has priority
over the other; simply
work from left to right.
Mnemonic device: Please
Excuse My Dear Aunt Sally.
Small group work: Students are
split into groups. Each will
work on a problem and present
to the class while explaining the
reasoning and using the steps to
solve the problem.
Problems:
a) 4 + 5 * 6 =?
b) 3 * 10 / 5 + 18 /3 =?
c) (4 + 5) * (2 + 3) –
(10*2) =?
d) (5 + 5) ^2
e) 5ଶ ∗ 3ଶ =?
f) 10ଶ + ሺ3 ∗ 8ሻሻ − 14 ∗
2=?
Week: Mar 18 - 22
Discussion of answers.
Wrap Up
Exit slip: Explain how you
used parentheses in Problem 6
on journal page 220 to write
the expression for the total
number of undamaged cans.
Students are making adequate
progress if they refer to the use
of nested parentheses to
identify the total number of
undamaged cans.
SL 7.4
Discussion of answers.
MJ: 223 and SL 7.5.
Math Masters pg 198
(Photocopy).
Homework
** Will be provided with breaks or additional time to complete assessment, if necessary.
Groups sharing problems.