ALIGNING THIRD-GRADE FRACTIONS CURRICULUM WITH THE COMMON
CORE STATE STANDARDS: A TEACHER’S PLAN FOR IMPLEMENTATION
A Project
Presented to the faculty of Department of Graduate & Professional Studies in Education
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF ARTS
in
Education
(Curriculum and Instruction)
by
Brooke L.N. Webster
SUMMER
2013
© 2013
Brooke L.N. Webster
ALL RIGHTS RESERVED
ii
ALIGNING THIRD-GRADE FRACTIONS CURRICULUM WITH THE COMMON
CORE STATE STANDARDS: A TEACHER’S PLAN FOR IMPLEMENTATION
A Project
by
Brooke L.N. Webster
Approved by:
__________________________________, Committee Chair
Stephanie Biagetti, Ph.D.
Date
iii
Student: Brooke L.N. Webster
I certify that this student has met the requirements for format contained in the University
format manual, and that this project is suitable for shelving in the Library and credit is to
be awarded for the project.
, Department Chair
Susan Heredia, Ph.D.
Date
Graduate and Professional Studies in Education
iv
Abstract
of
ALIGNING THIRD-GRADE FRACTIONS CURRICULUM WITH THE COMMON
CORE STATE STANDARDS: A TEACHER’S PLAN FOR IMPLEMENTATION
by
Brooke L.N. Webster
Statement of Problem
The last implementation of state standards was in 1997, when California, and the
rest of the country were in an economic climate that supported growth in so many aspects
including, education. In 2010, the state of California adopted new State Standards set
forth by the federal and state governments. California teachers are tasked with
implementing these standards; however, many teachers are not being provided with the
necessary tools to support their teaching. With larger class sizes, and little to no money,
teachers are being asked to change in an environment that offers them little to no support.
Purpose of Project
Common Core Standards are of the educational future in California. Due to the
lack of financial support in buying new curriculum that aligns to these standards, teachers
will have rely on their own professional practices to help them implement to these new
standards. For the teachers in many districts this means developing support lessons and
materials that align with these new standards while using their old non-aligned
curriculum as the base. This master’s project will help teachers to bridge the curricular
v
gap in the fractions unit of the third grade math curriculum, Macmillan/McGraw Hill
2008. The project’s supplemental lessons will coincide with the textbook lessons while
at the same time aligning with the CCSS.
Project Description
This project includes written lesson plans that can be easily implemented by third
grade teachers in my district and will assist them to be teaching based upon the new
CCSS. Using the current third grade curriculum alongside the Common Core State
Standards (National Governor’s Association Center for Best Practices and Council of
Chief State School Officers [NGA & CCSSO], 2010) as well as other fractions resources,
I developed supplemental fractions lessons that any teacher could use. In addition to the
detailed lessons themselves, the project contains all instructional materials including
student in-class worksheets and suggested assessments.
, Committee Chair
Stephanie Biagetti, Ph.D.
Date
vi
DEDICATION
This project is dedicated to all the math students in the state of California who were told
by their teachers that they would never be able to learn mathematical concepts, that they
were not smart enough to go to college, or even pass a math class. Those are the
individuals who have inspired me to make myself a better teacher of mathematics. It is
also because of my math teachers through my Jr. High and High School years that I
wanted to do this. I have proved them wrong. A student with dyslexia and who grew up
with little home life and in a single-parent household can be a learner and be successful.
vii
ACKNOWLEDGMENTS
I would like to acknowledge my amazing supervisor, Dr. Stephanie Biagetti.
Without her, this project would not have made it to completion. She gave me her
precious time, honest comments, and unwavering support throughout the course of my
writing.
I would also like to acknowledge my amazing partner and support, my husband
James. He has always shown me unconditional love. His unyielding financial,
emotional, and educational support has kept me inspired to see this project to completion.
I also need to thank my two children, Alexis and Ryan. They always supported
“mommy” with kisses and good wishes as I left to be in class. Without knowing it, they
sacrificed parts of their childhood so I could be successful. The greatest lessons I have
learned through this master’s program were because of the three of them. Lastly, I need
to thank my father. He raised me by himself and instilled in me the need to persevere
when the odds were against me. He did not live long enough to edit my final
masterpiece; however, it exists because of him.
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TABLE OF CONTENTS
Page
Dedication ......................................................................................................................... vii
Acknowledgments............................................................................................................ viii
Chapter
1.
INTRODUCTION ........................................................................................................1
Statement of the Problem ......................................................................................1
Significance of the Project ....................................................................................4
Research Questions and Anticipated Outcomes ...................................................6
Definition of Relevant Terms ...............................................................................6
Description of the Innovation/Intervention ...........................................................7
Limitations ............................................................................................................9
2.
REVIEW OF RELATED LITERATURE .................................................................10
The History of Curricular Change ......................................................................10
Why the Focus on Common Core Standards? ....................................................12
Emerging Education Issues and History .............................................................13
Reasons for Needing Supplemental Mathematics ..............................................15
Summary .............................................................................................................17
3.
METHODOLOGY .....................................................................................................19
Introduction .........................................................................................................19
Selecting a Topic for the Supplemental Lessons ................................................19
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Gathering Information to Develop the Supplemental Lessons ...........................20
Writing the Supplemental Fractions Lessons .....................................................21
Summary .............................................................................................................23
4.
DISCUSSION, CONCLUSIONS, LIMITATIONS, AND
RECOMMENDATIONS ...........................................................................................24
Discussion ...........................................................................................................24
Conclusions .........................................................................................................29
Limitations of the Project....................................................................................30
Recommendations ...............................................................................................31
Reflections ..........................................................................................................32
Appendix. Supplemental Lessons ......................................................................................33
References ..........................................................................................................................91
x
1
Chapter 1
INTRODUCTION
The Common Core Standards have prompted a curriculum change throughout the
United States; however, California does not have the funds to see this change through.
With these changes has come the angst of not having the supplies or curriculum teachers
will need to be effective in the classroom. Being a classroom teacher, I have an
understanding of what is needed by teachers to implement the new standards. We need to
create our own supplemental materials until the state is willing to provide us with the
curriculum. My project offers support for the fractions unit of my district’s current
curriculum, until the state provides its teachers with a mathematics curriculum.
Statement of the Problem
California recently adopted the Common Core State Standards (Fensterwald,
2012). The last implementation of new math standards, California Mathematics
Standards, was 15 years ago in 1997. The economic climate at that time was such that
teachers were supported by extensive and ongoing professional development to support
the curricular and instructional change. At the time of this study, a whole “generation” of
teachers had not experienced such a dramatic curricular shift. The most substantial
changes the teachers made most likely occurred when districts adopted new math
textbooks or implemented the use of new technology. Even modifications such as those
were accompanied by district-supported professional development, school-based math
coaches, and time for collaborative cooperation, resources scarcely available in the
2
current economic climate. Now class sizes are larger, school years are shorter,
expectations are higher, and teacher accountability has increased. Teachers today are
faced with tougher teaching circumstances and little to no support to achieve them.
The purpose of my master’s project was to develop a fractions unit to supplement
a third-grade math curriculum, Macmillan/McGraw Hill 2008, while incorporating the
newly adopted Common Core State Standards (CCSS) (National Governor’s Association
Center for Best Practices and Council of Chief State School Officers [NGA & CCSSO],
2010). With the lack of funding at the state and district levels for professional
development, teachers may have a difficult time correlating their current curriculum and
learning activities to the CCSS prior to the release and adoption of the new textbooks that
will align with the CCSS. Given the current budget crisis, I do not foresee adequate time,
resources, and compensation for the type of instructional implementation required for
students to achieve the CCSS and be adequately prepared for the upcoming CCSS
assessment by The Smarter Balanced Consortium (Ross, 2010). For my project, I
developed comprehensive math lessons that can help third-grade educators bridge the gap
between the current district-adopted curriculum providing a more procedural focus and
the new state fractions standards that take a deeper, conceptual approach through
reasoning and justification.
My concern with the current district-adopted mathematics curriculum is that it
was not developed with the CCSS as its guideline. Now that the state has adopted the
CCSS, teachers are left to navigate lesson preparation so they can support the new
3
standards without being provided the necessary resources, due to the unprecedented
budget crisis, to ensure success. Consequently, teachers have new standards to teach and
inadequate support to transform the current curriculum, assessments, and teaching
strategies to deliver them. By developing lessons to support teachers, I provided a
concrete foundation for the teachers to move forward in their own practice to develop and
implement lessons. Such a foundation makes it possible for teachers to teach the
curriculum and ensure they are adhering to the CCSS requirements, while at the same
time facilitating their students’ development of deep, conceptual fractions knowledge and
preparing them for the CCSS assessments that most likely will be as high-stakes as the
current CST exams.
As an educator, I find this problem significant enough that the development of a
supplemental unit is required, allowing teachers to focus their attention on teaching a
particular concept using premade curricular materials rather than being overwhelmed
with adapting the current curriculum or creating their own curriculum to ensure they are
teaching solely to new standards. Teachers lack the financial support to pay for
professional development or new curriculum; I hope to provide a basis for teaching key
math concepts and support my colleagues with extension lessons. Teachers will be held
accountable for the implementation of the new CCSS in 2013 with the first
implementation of the assessments during the 2014-2015 school year. By developing a
series of lessons in the area of fractions, I hope to make the transition for this area of
learning cohesive for the teacher and the student.
4
Significance of the Project
Teaching the CCSS is now required in the classrooms of the majority of
California districts. The standards have been adopted by 45 states of the union, one of
the more recent ones being California. The evidence is strong enough to indicate the
CCSS will not be ignored anytime soon by the state or school districts. Teachers need
curricular resources to guide their practice during the time gap between the
implementation of the CCSS and the adoption of the newly aligned mathematics
curriculum. My goal is that the project will address this time gap and provide support for
colleagues’ fractions math lessons in the interim.
After conducting an informal survey of teachers, I found there were still quite a
few colleagues who had little to no knowledge of the Common Core State Standards for
math. This was alarming to me. I felt educators have a certain level of responsibility to
make sure they are aware of topics on the forefront of education. The Common Core
State Standards (NGA & CCSSO, 2010) are one of those topics; they will change the face
of education for the coming years due to their monumental shift in focus from
mathematical procedural acuity to a more balanced approach using concepts, procedures,
applications, and mathematical reasoning.
The problem with this circumstance, as well as many others in education, is that
districts are faced with little money to support what the State government is asking of
their teachers. Curricular materials are just now being developed, but districts have few,
if any, resources to purchase the required materials to align the current curriculum with
5
the new CCSS. Therefore, teachers are left to work with their current misaligned
curriculum and make sure the new standards are being met. As a practicing teacher
myself, my situation is no different. I will need resources to aid me in teaching the same
third grade math concepts I have been teaching for years. The difference will be that
curricula must be aligned with new standards because I will be held accountable for my
students’ learning of concepts outlined in the CCSS. A main objective for the
development of these comprehensive and connected support lessons for third-grade
fraction standards is for me to implement them in my own classroom in the future and be
able to share them with my school and district colleagues to use in their own classrooms.
Research has shown (Lewin, 2010) that typically, elementary level teachers
struggle with deep mathematical content knowledge, especially in fractions. Several of
the “shifts” required for teachers to transition from the current standards to the CCSS
require teachers to go beyond “how to get the answer” so they can approach fractions
from multiple perspectives and enable students to understand the numerous connections
among fraction concepts. Then they can prompt students to write and speak about the
connections through justifications. With the detailed curriculum I created, teachers will
not only be able to implement it, because the project will include detailed lesson plans
including possible questions and suggested formal and informal assessments, but I
believe it will also help teachers themselves understand fractions more deeply.
6
Research Questions and Anticipated Outcomes
The goal of my project is to provide teachers with a comprehensive set of
connected fractions lessons (a unit of study) at the third grade level that are aligned with
the new CCSS. The project is designed to address the question:
How can third grade teachers be supported through the curricular implementation of the
new Common Core State Standards in fractions while utilizing mathematics curricular
(Macmillan/McGraw Hill, 2008) materials aligned with the previous standards?
I address the above question by developing a series of connected lesson plans that
comprise a unit of study and related supplemental curricular materials in-service teachers
can utilize to bridge the gap between their current mathematics textbooks and the new
CCSS. I anticipate the lesson plans are user-friendly enough for teachers either to
implement in their entirety or to adapt to meet the unique needs of their students.
Definition of Relevant Terms
Common Core State Standards
Standards that have been adopted by many states to bring cohesiveness as a
country to our school curriculum (NGA & CCSSO, 2010).
Comprehensive Lessons
These are lesson plans that contain detailed descriptions of all learning activities
and that take all types of learners into consideration. They support the learner that
needs to “wiggle,” the student that needs to have minimal distractions, the student
that needs to be utilizing two or more modalities of teaching, etc. Comprehensive
7
lessons also include such details as instructional objectives, suggestions for
assessments (formal and informal) materials for the lesson, etc.
Connected Lessons
Lessons that draw upon previously learned mathematics concepts and link to
upcoming mathematics concepts.
Differentiated Instruction
Teaching strategies tailored to meet the different learning modalities of all
students (Clearinghouse, 2009). It includes many different aspects, such as the
way a lesson is presented and how students are assessed for understanding, etc.
Informal Assessments
A procedure for obtaining information that can be used to make judgments about
children’s learning behavior and characteristics or programs using means other
than standardized instruments (Powers & Gamble, 2012), such as observations,
checklists, short, written assessments.
Supplemental Curricular Materials
Materials a teacher uses to help support the lessons/concepts being taught and/or
reinforced.
Description of the Innovation/Intervention
Using my current math curriculum teacher’s editions, supplemental materials, and
the CCSS third grade fraction standards as my guide, I constructed comprehensive,
connected lessons to fall in line with the lessons in the textbook (if applicable) and the
8
concepts described in the CCSS. Because the goal was to create a series of user-friendly
lessons, the lessons are kept in the order of the current textbook as much as possible as
long as the order makes mathematical and conceptual sense. The curriculum should also
to be a useful tool for teachers across the school district. As such, the lessons are
organized in a manner that will make it easy for teachers to duplicate the work in a way
that is meaningful to them and their teaching. For example, each lesson includes a
detailed lesson plan that includes basic lesson plan fundamentals. Anticipatory sets,
objectives, formal and informal assessments, etc. were thoughtfully addressed. All
student handouts and master copies for the students are included. I also included the new
CCSS that each lesson addresses and how it can be correlated to our former state
standards so how everything is paired together is very clear.
The bulk of the project entailed writing detailed lesson plans teachers could use in
their entirety and develop supplementary curricular materials so teachers could easily
implement the learning activities described within the lesson plans. I developed my own
lesson plan template from the one currently used by the Sacramento State Multiple
Subjects’ Teacher Preparation Program, the one I used in my own Teacher Preparation
Program, and included areas I believe are important to support teachers’ best practices in
the classroom. The template includes a listing of the standards addressed by the lesson,
both content and academic language objectives for the lesson, assessments aligned with
the objectives so teachers can gauge the students’ progress toward the objectives,
required materials for the lesson, a detailed description of all the learning activities for
9
the lesson including key questions teachers could ask throughout the lesson, suggestions
for the differentiated instruction for various groups of students, and ideas to make
connections between prior knowledge as well as expected new learning outcomes. The
lessons plans contain enough detail so teachers will be able to implement the lesson to
produce the expected student outcomes, but they also leave enough leeway for the
teachers to adjust the lesson to meet the specific needs of their own students. In addition
to the lesson plans, I created all supplemental materials including activity sheets,
worksheets, and any other materials required to successfully follow through with the
learning activities.
Limitations
There were several limitations to this project. The content of the project spans
only one chapter of the third grade mathematics curriculum utilized by the FolsomCordova Unified School District (Macmillan/McGraw Hill, 2008). It does not cover the
other 13 chapters in the third grade curriculum. Consequently, it leaves teachers to
develop their own adaptations to the current textbook for the bulk of the third-grade
curriculum. Moreover, it may not be useful to any other grade level due to the fact I am
supplementing the third-grade fractions curriculum. However, third-grade curriculum
has adopted CCSS and it varies little from district to district for direct instruction, only
for remedial instruction. In addition, the process I utilized to create the fractions unit can
be replicated by any teacher in any district, in any state in the country, thus reducing
limitations.
10
Chapter 2
REVIEW OF RELATED LITERATURE
This chapter contains a review of relevant literature that supports the need to
develop curriculum aligning current math curriculum with the new Common Core State
Standards (NGA & CCSSO, 2010). Specifically, the history of curricular change in
California mathematics, the support for teachers who accompanied the changes, and the
outcomes of the changes was explored. Also, the Common Core Standards movement
itself in terms of its roots, why it was necessary to have a national set of standards, why
California chose to adopt the standards, and the best practices for the most effective ways
to support teachers through the mathematics curricular change, primarily focusing on
teaching fractions with understanding was studied. In the end it was important to
articulate the most effective ways to support teachers through mathematic curricular
change. By developing a Common Core aligned fractions unit for the classroom; this
project will help teachers make a smooth and balanced shift to Common Core Standards.
The History of Curricular Change
Education in California is a process of constant change, some good and some bad
(Wu, 2011). With the passage of Goals 2000: Educate America Act by U.S. Lawmakers
in 1994, the country recognized the importance of the push for high standards to help
improve this nation’s education system (Pattison & Berkas, 2000). This push filtered
through the states, school districts, administrators and, finally, into the classrooms.
Along the way, implementation varied widely (Fensterwald, 2012).
11
In the 1990s, the state of California saw a push for standards-based learning and
teaching (Powers & Gamble, 2012). Having high academic standards is not enough if the
standards themselves are not implemented through powerful instructional methods
(Mooney & Mausach, 2008). There was a plan for developing curriculum framework
(Sawchuk, 2010). State frameworks were originally written by the State Department of
Education (California Department of Education [CDE], 2013). School districts then
developed their own standards-based curriculum to be implemented throughout their
school districts (Pattison & Berkas, 2000).
In 1994, U.S. lawmakers recognized the importance of high standards to help
improve education (Pattison & Berkas, 2000). Having high standards means having
qualifying educators to deliver the materials to make teachers and students successful
with curriculum and learning within the classroom (Hightower, 2011). As a current
teacher, this researcher understands supplemental curriculum is one way of supporting
teachers in the classroom, giving them the flexibility to enhance the current curriculum
that they are required to teach either due to school district or state decisions.
Supplemental material is also a progressive way of enhancing existing curriculum,
especially when funds are not available for school districts to purchase the necessary
curriculum (Carson, 2011). As California teachers move toward new Common Core
Standards, with no materials being provided, supplementing what they currently have is
going to be a vital tool in helping them teach. Considering California is only providing
professional learning modules (PLMs), basically online courses, teachers will have to
12
figure out how to implement these into their lessons or make their own lesson plans based
on what they have learned through these PLMs (Clearinghouse, 2010).
Why the Focus on Common Core Standards?
In the early part of the decade, there were many discussions about the test scores
in this country and what could be done about them. The George W. Bush administration
still felt there was some validity to NCLB, enacted in 2002 (Jennings, 2009). However,
when the Obama administration took over, it was determined this country needed more
continuity in the student content standards across the country; thus, the Common Core
Standards discussion began (Lewin, 2010). Understanding that uniformity of curricular
standards was needed within the United States to help make the country more successful
in the global community discussions about a common set of standards began to take hold.
With all states maintaining their own individual curriculum standards, this country’s
achievement gap had hit an all-time high. It was time to focus on continuity, and having
universal state standards was how the country was going to achieve it.
According to Tamar Lewin (2010), adoption of the standards “will not bring
immediate change in the classroom. Implementation will be a long-term process, as
states rethink their teacher training, textbooks, and testing” (p. 2). By school districts
focusing on Common Core State Standards, the education community strives for
outcomes more uniform with student learning. Working teachers will understand exactly
what common set of content standards our students need to know to be successful at the
college level (Lewin, 2010).
13
Implementation of the Common Core Standards will require a systematic
approach to implementation. Reeves (2010) laid out a realistic process for putting the
standards into practice. The first step will be to find common ground and reassure
teachers that many of the Common Core Standards are already incorporated into their
current mathematics curriculum. This master’s project delineates the “common ground”
and also provides support to address the Common Core Standards not in the current math
curriculum.
To meet the needs of all students, including those with special needs, teachers
need to understand and be prepared to implement Common Core Standards
systematically by focusing on one concept in depth before moving into the next
(Gresham & Little, 2012). In addition, teachers will need to embrace common formative
assessments versus just one type of end-of-year grading assessment and understand how
to use the formative assessments to guide their practice (Ball, Hill, & Bass, 2005).
Finally, using the standards as a foundation for best teaching practices will aid teachers in
reaching successful student learning outcomes. The Common Core State Standards do
not need to be a standardized form of teaching but more a creative outlet for letting
accomplished educational professionals do what they do best: effectively teach (Wu,
2011).
Emerging Education Issues and History
Throughout history, changes in education have been brought on by changes in our
nation’s economy and work place. “Recommendations for educational innovations
14
brought on by the issues and events of nineteenth century industrial America included
new types of school, new curricula and methods of measuring success of programs”
(Ross, 2010, p. 1). It is this consistent need and type of change that helped bring forth the
discussion and thought process for the development of the Common Core Standards.
Ross (2010) continued to suggest, “major social and economic factors cause
tremendous changes in education.” It would explain why early into the 21st century it
was deemed necessary to focus on how to make education balanced with regard to the
needs of the students of the United States. The government was changing, the United
States was in the thick of the worst economic recession this country has seen since the
Great Depression, and state test scores were still dismal and showing little to no growth
for consecutive years (Ross, 2010). Consequently, the federal government began its
investigation of what kind of curriculum and standards could improve education (Melton,
2011).
The process of adopting new curricular standards had numerous steps. The year
2009 was significant for implementation. It was the year in which 48 states, including
California, committed to developing a set of standards that would help prepare students
with the knowledge and skills needed to succeed in education and careers after high
school. In January 2010, Senate Bill 1 (SB X5 1) established the Academic Content
Standards Commission in California to start making the necessary recommendations to
the Governor in regard to approving or disapproving the CCSS (Wu, 2011). By August
2010, the California State Board of Education voted to adopt the CCSS. Apparently at
15
some point, there was a brief time for public comment; however, as a state educator, this
researcher does not recall anything ever trickling down. Two years later in August 2012,
the Folsom Cordova Unified School District, Kindergarten and first grade, began to
implement the standards, and in the next two years, the rest of the elementary grades
followed. Unfortunately, no money is being put forth by the state government to support
districts and teachers in implementing these new standards until 2015 (Powers &
Gamble, 2012). Until then, professionals are left to their own devices and knowledge to
incorporate these standards using their best practices by means of assessments and
teaching practices (Powers & Gamble).
Reasons for Needing Supplemental Mathematics
Understanding that neither the federal government nor the state government
currently has adequate funding to support America’s teachers in the implementation of a
CCSS-based curriculum has led to heated debates and discussions. On September 25,
2011, President Obama addressed the current woes of the country’s education system.
He discussed the amount of money per pupil we put into our students, which according to
the interviewer was “comparable” with other first-world nations, yet we are still not able
to achieve any major success. He replied with “My sister was a teacher, I am fully aware
of how often the conscientious teacher pulls money out of her own pocket to put into her
own classroom and how taxing it can be for the teacher” (Jennings, 2012, p. 1). Yet,
once again, as a nation, teachers are embroiled in a great change to education and without
adequate funding for them. Personally, after being in the classroom for 15 years, I am
16
fully capable of taking a lesson and “morphing” it to fit the needs of my students, but
others may not be so willing and able (Jennings, 2012).
History Proves the Need for Math Specialist to Support Mathematics Curriculum
During the 1960s and 1970s, “departmentalizing” (Fennell, 2011, p. 53) was
proposed at the elementary level. Educational enthusiast felt that the students having
access to teachers that specialized in one academic area was what children needed to
excel. According to the NCTM President at the time there was some validity to that
departmentalizing at the elementary level. In an article authored by him, he made a
passionate plea for mathematical specialist in elementary schools (Fennell, 2011) because
they could ensure children were getting what they needed in the field of math.
By the 1990s, school districts had mathematics “lead teachers” whose role was to
give continual training to classroom teachers about changing methods, curriculum, and
standards. The primary focus for these “specialists” (Fennell, 2011, pp. 53-54) was to
keep all current classroom teachers apprised of what students needed in terms of math.
New teachers began going through Beginning Teachers Support and Assessment (BTSA)
and were given the support they needed to develop their own best practice (Melton,
2011). Seasoned teachers were given the opportunity to focus on their skills and refine
them. Schools had money, strong curriculum, and a community that supported this
moving forward action. Now they have much less.
By 2008, the financial crisis was taking a stronghold across the country and
education was taking a financial hit of unprecedented levels. Teachers were being laid
17
off by the hundreds and more and more lead teachers across districts were pushed back
into the classroom. Thus, teachers no longer had district professionals to count on for
support in mathematics (Melton, 2011). With the adoption of the Common Core
Standards, teachers are now faced with another major shift in their curriculum and
expected student learning outcomes (Sawchuk, 2010). Math Specialists have the
knowledge to keep instrumental concepts and teaching standards from being forgotten, or
set aside altogether through the implementation of CCSS. The curriculum on which this
project focused is intended to help bridge that gap for the teachers and give them the
supplemental lessons they need to make them more effective in the area of fractions,
while implementing CCSS and using the materials they have.
Summary
The question truly is not why curriculum is needed to support teachers during this
trying time in education. The real question is how teachers will be supported through
implementation of CCSS. This project addressed this question. Various authors and
educators have numerous examples as to why this important move needs to happen now.
Throughout this literature review, various arguments and examples have been given that
make it clear as to why teachers need this curricular assistance. Without the money,
teacher training, and curriculum aligned with the new CCSS, teachers are going to be left
to their own professional mechanisms to make their math lessons successful and
compliant. By developing support lessons in fractions for third grade teachers, teachers
18
are provided with the tools to ensure their own fraction lessons are aligned with the
CCSS.
19
Chapter 3
METHODOLOGY
Introduction
Understanding that teachers come from all areas of the world and with varying
levels and types of educations, the vast majority of starting teachers did not acquire the
necessary understanding of K-12 mathematics while in college (Wu, 2011). With the
Common Core State Standards being implemented across California, I saw an
opportunity to broaden my own perspective on specific mathematical concepts and gain
the knowledge I would need to successfully implement the new CCSS. Through personal
experience with teaching fractional concepts and a deficiency of conceptual
understanding of fractions, I found an opportunity to grow as a professional and develop
supplemental math lessons that could support other elementary teachers. By developing
these supplemental lessons, it was hoped other teachers would be able to deepen their
own understanding of the fractional concepts through their implementation of the lessons.
Selecting a Topic for the Supplemental Lessons
Developing supplemental lessons would be the most significant piece to the
project, with a goal of constructing thoughtful lessons that not only would be easy to
implement but that would also support teachers at all levels in their career, from the new
teacher to the seasoned teacher. When narrowing down the mathematics topic on which
to focus the project, a topic that had been difficult in my own teaching and, more
importantly, difficult for the students to learn and understand at a deep level, was
20
selected. Consistently, learning fractions had been a painful experience for my students;
conceptually, some students had tremendous difficulty grasping even the basic
fundamentals, and my concern was exacerbated by the fact that many future
mathematical concepts and topics rely on fundamental knowledge of and computation
with fractions. On a personal level, fractions had always been a difficult concept for me
to understand. As such, it was determined I would develop supplemental fractions
lessons that would not only empower teachers in the classroom but then would improve
student learning outcomes for fractions.
Gathering Information to Develop the Supplemental Lessons
As part of my need to understand what new teachers might require in
supplemental lessons for fractions, I decided to attend several sessions of an Elementary
Mathematics Methods course. These classes were held at Sacramento State University,
were part of the Multiple Subjects Teachers Preparation Program, and focused on a
multitude of fractions concepts. When auditing this class, I was intrigued by some of the
candidates’ lack of mathematical knowledge and understanding they needed to
effectively implement the new mathematics Common Core State Standards and utilize
best practices. I recognized the candidates were just beginning their teaching preparation
program, but I also realized the mathematics Common Core State Standards require a
more comprehensive understanding of foundational concepts than the previous standards
required. As a result of my observation and participation in the mathematics methods
course, I decided to include, at the beginning of each lesson, a section devoted to a
21
detailed explanation of the lesson purpose and the role of the content in building the
students’ foundational knowledge of fractions, thereby educating the reader at the same
time.
The next step in developing the supplemental lessons was to study in depth the
new content standards for third-grade fractions as well as the Standards of Mathematical
Practice, which proved to be the greatest departure from the old state standards. When
referring back to my own curriculum (Macmillan/McGraw Hill, 2008), it quickly became
apparent to me several standards in the mathematics Common Core State Standards were
not even addressed by the current curriculum. As such, I planned to develop a separate
supplemental lesson plan that would correlate with each of the fractions Common Core
standards and create or find activities and materials that would support each lesson.
Consequently, I went to teacher stores to obtain manipulatives and teacher support books
appropriate for the new fractions content standards. In addition, I searched the Internet
for lessons, activities, and resources aligned with the Common Core third-grade fractions
standards. A content analysis was used to select the activities and materials I needed.
Using mathematics Common Core content standards and the Standards of Mathematical
Practice as a guide, I selected the materials in the next section as resources for the thirdgrade supplemental fractions lessons.
Writing the Supplemental Fractions Lessons
The goal of the supplemental lessons is to provide teachers with comprehensive
lessons that include a variety of in-class and independent learning activities as well as
22
daily informal assessments and a summative formal assessment. Each supplemental
lesson targets one fractions Common Core standard at the third grade level. The lessons
begin with an explanation of the purpose of the lesson and how the content and activities
contribute to a fundamental conceptual understanding of fractions. The Common Core
standard addressed by the lesson is also listed for quick reference as well as the
instructional objectives and the materials required for the lesson. Anticipatory set, main
learning, and closure activities are suggested including possible questions teachers can
ask throughout the lesson. Finally, accommodations are suggested for students with
special needs, English language learners, and advanced learners.
Each “Ticket out the Door” (end-of-lesson assessment) was created to align with
the learning activities within each lesson as well as the Standards of Mathematical
Practice. The summative formal assessment was adapted from the curriculum guide
Common Core Mathematics and the content was drawn from each individual lesson. I
chose to include two different assessments. The first one is more traditional in nature as
it is in a selected-response format. The second assessment is more aligned with the
CCSS expectations as well as the expectations for the Standards of Mathematical practice
and provides the opportunity for students to show a deeper understanding of fractions
concepts. In addition, the students are expected to explain their mathematical thinking in
more depth. In giving two examples, I feel I have given teachers the opportunity to use
their professional judgment in deciding which model best fits the needs of an individual
student or the class as whole. Finally, it was important to organize the lessons in a
23
logical manner. Using the current curriculum was not an option, as it does not follow the
order of the CCSS. Consequently, I chose to follow the order of the published standards
so teachers would be able to easily match the sequence of the lessons to the standards.
Once a teacher feels more confident in the teaching of the new standards, it is assumed
she or he will make the professionals adjustments she or he deems necessary to
effectively teach third-grade fractions.
Summary
This curriculum was created around the Common Core State Standards with a
teacher’s immediate needs in mind, having little support to implement the actual teaching
of the new fraction concepts and standards. Throughout the development of the
curriculum, I focused on making sure the lessons were aligned with CCSS and Standards
of Mathematical Practice adopted by the state of California. This curriculum is designed
for teachers in need of supplemental material at any point in their career, with a hands-on
approach to ensure they are implementing the new CCSS for third-grade mathematical
fractions.
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Chapter 4
DISCUSSION, CONCLUSIONS, LIMITATIONS, AND RECOMMENDATIONS
Discussion
Teaching and learning standards have been an intricate part of education. For
decades, these standards have been under constant metamorphoses and growth.
Standards-based curricula are the basis for how teachers develop their lessons and the
delivery of those lessons. They are a guideline for the learning objectives and outcomes
of all that takes place in a classroom. Without standards, the modern classroom and
teacher would not have the cohesiveness education needs to be efficient.
When the Obama administration took office in 2008, they made it very clear that
changes would be coming to the national education infrastructure. With the lack of
support for the “Race to the Top,” the administration knew they had to switch strategies
and develop another way to unify education in the United States; this started the
discussion for developing a set of nationwide standards that would give school districts
an option of being aligned as a nation (Herbert, 2011).
In 2010, California adopted the Common Core State Standards. In implementing
the standards, districts have been granted the flexibility to add up to 15% supplemental
standards to address the needs of their own students (CDE, n.d.). This allows for local
flexibility while still enabling districts to participate in the national movement for a more
consistent educational experience for students. Even though the state has added to the
25
standards to fit our state’s needs (CDE, n.d.), we are still faced with implementing new
standards and limited materials to support the shifts on our classrooms.
Educators are faced with the realization that they need to adjust their teaching and
lesson planning with a professional level of creativity and adherence to the standards.
However, with that understanding, teachers have generally been provided with some sort
of district support and/or materials to make sure standards were being taught in the
classroom. With this “shift” in standards this is no longer the case; teachers will now be
required to adjust their instruction with little district support and few supporting
materials. This lack of support was what prompted an exploration into writing CCSS
lessons for the third grade fractions unit in my district.
Determining the Importance of Each Lesson
Considering the different aspects of the standards, as well as best practices, and
comparing them to what was present in the existing Folsom-Cordova Unified School
District curriculum, the lessons were constructed to allow for a variety of mathematical
experiences for the students. All the lessons have some level of a “hands-on” experience.
This experience could be as easy as constructing the manipulatives that help the students
complete a certain assignment (i.e., the fraction circles), or something as simple as
allowing the students to use a personal white board. Research supports that putting
something tangible into the students’ hands can help enrich an otherwise boring or
difficult concept (US Department of Education, 2010).
26
Design of the Unit
For simplicity of understanding, the lessons are designed using an easy-to-follow,
user-friendly format. Moreover, the lessons are presented following the order of the
standards the state published on their CCSS web site (CDE, n.d.). Since California has
yet to adopt curriculum aligned with the CCSS, the order of the fractions lessons has yet
to be widely determined. Even though the lessons in the unit are ordered in the same
manner laid out in the CCSS, teachers can use their professional judgment to order the
lessons as they see fit.
Colleague Input on the Unit
The input from my colleagues has been very supportive and helpful. I gave four
teachers at different elementary grade levels the same lessons with the same
questionnaire. Two of the teachers taught third grade, one colleague taught fourth grade,
and the last colleague taught at the middle school. I chose all the teachers who read my
lessons for different reasons. The two at my grade level will be teaching the new CCSS
next year and were very open to any material that could be used to help them make this
shift in their teaching. My colleague in the fourth grade was interested to see what her
future students would be learning and I was interested to know if what I was doing was
relevant and helpful in getting my students ready for the next grade. My colleague at the
middle school level is a CCSS coach, and has had the experience of more in-depth
training with the standards; therefore, I was interested in getting her feedback to make
sure I took my unit in the correct direction in terms of making it helpful to other teachers.
27
Of the four I surveyed, they all said that they found the lessons helpful and liked
the fact that differentiated suggestions were made and that each lesson had some sort of
“hands-on” activity or support to go with them. They also said they were laid out well,
but found them “lengthy” in terms of the amount of questioning. One teacher did not
care for the allotted time I suggested with each part of the lesson. Her comment was “I
felt there was too much information for me to make sure the lesson was completed in the
allotted time frame.” I provided time allotments to primarily help with new teachers who
often felt rushed to get the information delivered when teaching a new lesson. The time
allotments do not need to be followed, they are just guidelines to assist with teacher
planning.
One of the teachers commented on the fact that the vocabulary is consistent
throughout the lessons and is helpful to the students when retaining different pieces of the
fraction unit. I agree with this suggestion; I feel it leads to a consistency throughout
many lessons and can help students begin to understand how each fraction lesson can
build upon the other.
Another good suggestion from a colleague was to make a “cheat sheet” for
veteran teachers. These lessons were each multiple pages long and veteran teachers
should possess the skills to manage a lesson without having it completely “spelled” out to
them. This was a concern to me through the writing of the lessons. I never wanted any
colleague to feel I did not honor their own experience. However, I was writing to a wide
variety of skilled teachers from the beginning teacher to the veteran teacher. I thought a
28
“cheat sheet” would be a good addition to consider. By just highlighting the key
concepts and standards needing to be addressed, along with some optional teaching
concepts, it would allow the more seasoned teachers to implement the lesson in their own
fashion using their own techniques.
Professor’s Input on the Unit
Dr. Brian Lim was extremely helpful and supportive when agreeing to read and
give professional feedback on my fraction unit. He was easy to work with and offered
useful suggestions in a very helpful way. One of the suggestions Dr. Lim offered was in
regard to my assessments, both formal and informal. He felt they could be more specific
in terms of questioning and how they are presented in terms of where they are placed in
the unit, directly after the lesson or at the back of the unit. I tried to develop and locate a
few different types of assessments. The curriculum currently being taught in my district
is lacking new CCSS lessons. Teachers do not yet have support materials provided by
the district, and the district and the state do not have assessments to provide to us, either
informal or formal. At the time of this writing, my district does not know what we will
be administering in regard to any standardized assessment, including any type of state
testing in the spring of the 2014 school year. Therefore, we do not even know what type
of test to prepare our students for.
With no district-approved assessments, it is extremely difficult for me to design
assessments for this unit without an example of what is truly being expected as whole.
Understanding that assessments are one of many resources that should help a teacher
29
guide his or her instruction, it is important to have relevant assessments for the students
and teachers. However, when faced with the dilemma of teachers not aware of their
district’s standardized assessments, many teachers will be assessing purely for their own
information and data. I did like Dr. Lim’s suggestions of placing my provided
assessments with each lesson when applicable. I would hope that the “ticket out the
door” is used to quickly assess and help guide immediate instruction, especially while
teachers are still learning how to effectively implement the CCSS.
Another helpful suggestion was for me to be more specific when referring to
students with special needs and what my lessons provided them. There are many levels
of “special needs” students. Not all suggestions will fit their individual learning styles
and needs. A student with 504 plan requirements can be very different from what a
student with an IEP and a dyslexia diagnosis needs. The part of these suggestions I find
difficult in terms of this project is how to give these details without overwhelming a
teacher who is simply trying to teach a CCSS math lesson for the first time. Again, this
will be another area teachers will have to evaluate, incorporate into their own teaching
style, consider the needs of her students, and perhaps consult with a special educator to
know what will be most beneficial for student learning.
Conclusions
As a third grade teacher, I am happy about the recent “shift” we are seeing in our
state standards. Streamlining education and getting all educators on the same curricular
page is a good idea. However, I agree with SITE that with the lack of implementation
30
support and funding, the positive force behind the CCSS may fade quickly and once
again leave educators struggling to make sure every student is being taught to their fullest
capabilities.
According to Wurman (2012), Common Core Standards are fewer, clearer and
more demanding than many other state standards. Common Core standards were
developed for uniting education on a national front. Sawchuk (2010) mentioned the
“potential challenge” is to get teachers to change their teaching methods and thought
processes of “attempts to reshape their teacher training and crafting new methods” (pp. 12). Changing teachers is a difficult process, especially when there is little support for
teachers to make that change.
CCSS are here to stay for the immediate future. This set of standards may very
well be the last significant change educators see during their careers. With the
completion of this project, I hope I have made a significant change to my own teaching
practices in the classroom in regard to mathematics. If the material I created is helpful to
a colleague it would be an added bonus.
Limitations of the Project
As mentioned previously, there are several limitations to this project. The
specific content of my project is derived from just one unit of the third-grade
mathematics content rather than the entire year. It would have been impossible to tackle
the content of the entire text given the program’s time constraints. Therefore, teachers
must develop their own adaptions to their current textbook for the remainder of the third-
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grade curriculum. In addition I drew the content from the curriculum utilized by the
Folsom-Cordova Unified School District (Macmillan/McGraw Hill, 2008), which may be
different from other districts across the state and across the nation. As such, teachers
who does not use the same text need to adopt these lessons even more to implement this
project. Furthermore, the project may not address teachers’ fraction lessons needs for
any other grade level given that the lessons target only the third grade fractions CCSS.
However, in my opinion the lessons can be easily adapted to either the second or fourth
grade fractions CCSS if a teacher is willing to make the adjustments necessary for the
needs of their individual classroom.
Recommendations
The next part of this process is going to be implementing the unit in my own
classroom and allowing my immediate colleagues to attempt the lessons but in their own
classrooms. Once this happens, I will reflect on my personal implementation as well ask
my team how they feel the lessons went in their classrooms. I also think it would be
interesting to see if all the teachers on my team would teach the same lesson but a bit
differently. This would have to take place after one teacher taught it and then the
teachers discussed it and gave the other teachers suggestions to build upon. After the
debriefing among colleagues, the lesson can be taught again using the recommendations
the team made, thus mimicking an informal lesson study.
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Reflections
I started the Master’s program as a personal professional journey. My entire
educational career was about obtaining an education to make my father happy or to
establish myself within a profession. When it came time for me to select what my project
would be based on, I knew I wanted a topic that would help other educators, while at the
same time improve my own teaching skills. I had always been weak in math, which
flooded over into my mathematical teaching skills. By choosing to write a fraction unit
based on the new CCSS, I knew I was helping my fellow teachers, students, and making
myself a better elementary math teacher.
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APPENDIX
Supplemental Lessons
Fraction Circle
Purpose of the Lesson: The purpose of this lesson is to introduce students to unit
fractions and non-unit fraction by using a visual model – the fraction circle. For this
lesson, students will construct fraction circles, which are pie-shaped fraction tools that
can be broken down into halves, thirds, fourths, sixths and eighths. Students will use this
manipulative throughout the unit for fractions.
Common Core Standards: 3.NF 1. Understand a fraction 1/b as the quantity
formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as
the quantity formed by a parts of size 1/b.
Instructional Objective: By the end of this lesson students will be able to express unit
fractions and non-unit fractions using the visual model they created. They will also be
able to apply this knowledge independently to identify fractions given a visual problem
(e.g., shade 2/3 of this rectangle).
Assessment


Informal: Students will submit the worksheet that accompanies the fraction
wheel activities. This is also a good opportunity to have students work in small
groups, complete the worksheet together, and discuss what they are doing.
Responses to questions asked during whole group, small group, and independent
work will also indicate the level of the students’ understanding.
Formal: Provided assessment if applicable.
Materials


Have student’s fractions wheels prepared ahead of time (parent can easily do
this, just use copy template).
Scissors
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Procedures
Anticipatory Set 15 mins:
1.
Have students create their visual model by cutting out the fraction circles. Each
fraction circle should be photocopied on different colored paper for ease of
reference (e.g., fraction circle that show halves are copied onto yellow paper;
fraction circles that show thirds are copied onto green paper)
2. Introduce concept of denominator: Discuss what equal parts look like and mean
using language to which students connect: e.g., if you and your friend want to
share this cookie (circle shown) so it is “fair,” how can we make equal parts so
everybody gets their fair share? If students require additional examples, similar
questions can be asked about sharing among four students to create fourths. The
denominator represents the number of equal parts that make up the whole
3. Introduce concept of numerator: using same of idea of fair sharing, use circle
divided into fourths. If you were sharing this “cookie” with a friend and it was
already divided into 4 equal parts, how could we share it fairly? Each friend gets
2 of the equal parts
a. Show fraction 2/4 on circle so that 2 of the 4 equal pieces are shaded in.
Retell the story so the students see the 4 represents the number of equal
parts that make up the whole and the 2 represents the number of equal
parts that each friend gets to eat.
 Main Learning Activities 30 mins:
The teacher will start the lesson by using the example of ½. Ask the students to take the
pieces of their fraction circle they think will show ½. Then check in with the students
and make sure they have pulled out the correct colors (white-whole, halves-yellow). The
teacher should do this part whole class, and demonstrate it as such. The teacher will want
to make sure the students can refer to the denominator as two equal parts that make up a
whole, and the numerator indicates one of those equal parts.
1. After the students have had a few minutes to manipulate their fraction
circles and get an idea of what they represent, the teacher should start
leading them through examples. Have them show the teacher what 1/3
looks like, then ¼, then 1/6. When asking the students to explain how
they know they are correct, the teacher should be listening for student
explanations that refer to the denominator and numerators for what they
35

mean. For example, for 1/3 the students would explain that there are 3
equal parts in the whole and we are just using 1 of those pieces.
2. Next have the students work out harder examples, such as 2/3. When
looking at 2/3, the student must explain that the 3 tells that 3 equal parts
make-up the whole and the 2 means that they have 2 of those equal parts.
3. Depending on the class, the teacher may want to ask the students how this
fraction would be applicable if they were out in the “real world.”
4. The next step will be to have the students work in pairs. Challenge the
students to demonstrate 3/6 and 4/8 with their fractions circles. The
teacher is also going to want the students to once again demonstrate what
they understand a numerator and denominator to mean. The teacher could
also challenge the students.
5. Lastly, have the students work on a few independent problems with a
partner. Give them the examples of 3/8 and 5/6. Instruct them to ask their
partners the same questions you just asked them. A suggestion would be
to write a sentence frame on the board. Example: What does the
denominator represent? How do you know this?
Closure Activity 15 mins: Have the students review the fractions taught, pair
them up and have them quiz their partner on how to make ½, 2/3, 3/6, etc. with
their fraction circle.
Accommodations



Special Needs: Using a manipulative in this lesson will support your lower
students. The wheel is also in various colors giving visual cues.
Advanced: See if they can identify equivalent fractions (e.g., when working with
3/6, can they also see this as ½). Have student show 5/8 and explain in detail how
their fraction circle shows 5/8 while referring to the denominator and numerator.
ELL: If the teacher chooses to put the students in small groups, this allows the
students to use oral language to work through the worksheet. If not, the fraction
wheel is in different colors, which can support students with visual cues.
36
References
Fractions, Grade 3, Teacher Created Resources, 2011
Common Core Mathematics-Practice at 3 Levels; Levy, 2012
Blooms Taxonomy Questioning
37
38
39
40
41
Fractions on a Number Line
Purpose of the Lesson: Representing unit fractions on a number line. The purpose of
the lesson is so students understand how to partition a number line into equal parts to
represent fractions on a number line. Students will eventually label the tick marks as
fractions and use the tick marks to plot points on the number line. A number line is
another version of a length model for fractions.
Common Core Standards: 3.NF 2 understands a fraction as a number on the number
line; represent fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0
to 1 as the whole and partitioning it into b equal parts. Recognize that each part
has a size 1/b and that the endpoint of the part based at 0 locates the number 1/b
on the number line
Instructional Objective: Students will be able to represent unit fractions on a number
line, by partitioning the whole into equal parts using tick marks.
Assessment


Formal: Test using a number line and having the students represent simple
fractions on the number line. Teachers may use their curriculum assessment if
applicable.
Informal: Ticket out the Door. Half of a worksheet with three number lines
already drawn on it.
Materials




Pencil, blank white papers, ruler, markers
Sentence strips
Formal assessment worksheet
Ticket out the Door
Procedures

Anticipatory Set 10-15mins: *NOTE*When the students were in second grade,
based on the CCSS, they should have been exposed to what a number line is and
how it works. I recommend teachers quickly review what a number line is: e.g.,
42
how it continues indefinitely, how it is labeled with numbers, how the tick marks
are equally spaced apart, how it is relevant to the students’ lives and what it
means to them mathematically. Draw a basic number line with your main
teaching device (ELMO or overhead) and have the students tell you what they
know. Then go into the formal start of the lesson, “You can use a number line to
show fractions. Look at this number line. [Show a number line 0-1, with an
unlabeled tick mark at 1/2] How does the number line show equal parts
(response: the tick represents equal parts)? How many equal spaces are there?
How would you represent ½? 1/3? Etc.” To get the students to focus on the
unlabeled tick mark in the middle and how it partitions the whole into two equal
parts, the teacher could ask these possible questions:
1) How does the number line show equal parts?
2) How many equal parts are there?
3) How could ½ be represented?

Main Learning Activities 30 mins: Partitioning the whole into fourths: Use a
number line with endpoints at 0 and 1 and with unlabeled tick marks that partition
the whole into fourths.
1. Point out the 0 and 1 under the first and last tick points. “The number line
shows 1 whole. Each space is one equal part of the number line. Because this
number line is divided into four equal parts, we can say it shows fourths, or
quarters. The tick marks divide the whole into four equal parts.”
2. Continue with adding tick marks, so eighths are created. “The tick marks
divide the whole into how many equal parts?” What fractions have been
formed?”
3. Continue to practice with students on the number lines. Break the whole on
the number line into sixths.
4. Break the students into pairs. Challenge the students to partition the whole
into thirds. To guide the students, ask questions such as: if I want to make
thirds, how many equal parts should I create? Talk with your partner about
where the tick marks should be placed in order to create thirds. If you want to
continue to challenge the students ask them to portion the whole into sixths
either independently or with their partner.
43

Closure Activity 15 mins: After completing the lesson of partitioning the wholes
into halves, thirds, fourths, sixths, and eighths, have the students recreate the
fractions independently and then ask them a few closure questions.
Example: “In your own words can someone describe how you know where to
place your tick mark?” “What would happen if your mark was in the wrong
place?” “How would you know?”
Accommodations:



Special Needs: Have them start with already partitioned number lines versus
having to partition one themselves.
Advanced: Give them a number line with multiple ticks and push them to label
the tick marks on the number line 0 and 1, or further.
ELL: Have them verbally describe to the teacher or a peer what a number line is
and what it looks like and have them describe the partitioning process.
References
Fractions, Grade 3, Teacher Created Resources, 2011
Common Core Mathematics-Practice at 3 Levels; Levy, Jill; 2012
Blooms Taxonomy Questioning
Example for Ticket out the Door:
Name:___________________________Date_________________________________
Finish the number line. Represent halves or fourths:
0
1
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Number Line Lesson B
Purpose of the Lesson: To take the previous number line activity one step further by
now having the students place their own tick marks on a number line. The students
should also be able to recognize that the resulting interval has a size and its endpoints can
be located on a number line.
Common Core Standards 3NF-B Represent a fraction as a number line diagram by
marking off lengths 1/b from 0. Recognize that the resulting interval has size a/b and its
endpoint locates the number a/b on the number line.
Instructional Objective: By the end of the lesson, students will be able to locate
fractions and their endpoints on a number line independently, meaning they will not need
to have an already prepared number line. The students will be able to make the number
line, and they will be able to place the tick marks in the appropriate places.
Assessment


Formal: Formal curriculum assessment if applicable. If the teacher has the
students also make their own number lines, plotting various fractions, it is
recommended to collect these as well. Sentence strip number lines can be used as
support manipulatives in another lesson if the teacher chooses.
Informal: Support worksheet
Collect sentence strips and make sure they were done correctly. The
students may keep them and use them as a reference when they are
doing classwork with future lessons. The teacher will also monitor the
lesson and the conversations taking place among the students. The
teacher will check in with student groups and ask them a series of openended questions spanning Bloom Taxonomy to determine what the
students are finding easy or difficult with this lesson.
Materials


Sentence strip number lines
Support worksheet
45
Procedures


Anticipatory Set 10 mins: As a class, review their background knowledge on
number lines (e.g., purpose of the tick marks, how they are labeled on the bottom,
endpoints indicated on the number line), skip counting, and fractions (e.g., what
the numerator represents, what the denominator represents, fractions as equal
parts of whole, etc.). Also, refer back to the previous lesson taught when the
students were provided with pre-labeled number lines. On the teacher’s main
teaching device (ELMO, overhead, etc.), have the students talk the teacher
through making a number line. Make sure the students are using the proper
vocabulary when doing this exercise. For example, tick marks, plots, and
endpoints of the number line. This will help the teacher determine if they retained
the previous concept and can move forward in this support lesson.
Main Learning Activities up to 50 mins: Have an in-depth discussion about
how the denominator represents the number of equal lengths the “whole” is
partitioned into on the number line (most likely the “whole” will be the interval 01), while the numerator represents the number of lengths from O. Start by having
the students make predictions about where fractions should be placed on the
number line. The students will then be given their worksheet, and partnered if
necessary. To help the students with spacing and depending on the class as a
whole, a teacher may want to at least have number lines already prepared.
However, the students will be responsible for adding the remaining parts to the
number line. Example, “Can someone tell me where the endpoint belongs?”
“Can someone tell me where the ½ point tick mark would go?” “How do you
know it goes here?” “Let’s be sure to label this tick mark as ½.” Once again, it is
important students understand that the denominator indicates how many equal
parts the interval 0-1 is partitioned into (e.g., if the denominator is 8, then the
interval 0-1 is partitioned into 8 equal parts). The numerator counts how many
parts from 0 the fraction should be plotted (e.g., if the numerator is 3, then the
fractions should be plotted a distance of 3 parts from 0).
Next, the teacher will need to teach the students how to plot non-unit
fractions on a number line as well. Using examples of 2/3, ¾ and 3/8, guide the
students to also plot these tick marks on a number line. For consistency, using the
same questions will help the student to understand they are plotting similarly to
the previous fractions. “Can someone tell me where 2/3 would go on a number
line?” “How do you know you are placing the tick mark in the correct place?”
“Is there any way to check that we’ve placed the tick mark in the correct place?”
If students are struggling, then have the students label all the tick marks within the
entire interval 0-1. For example, if the denominator of the fraction is 8, then the
students should partition the interval 0-1 into 8 equal parts with the tick marks and
label the tick marks 1/8, 2/8, 3/8, 4/8, etc.
46

Closure Activity 10 min: Start by summarizing both of the lessons. “What have
you learned about number lines?” “Why are they important?” “How are they
useful in plotting fractions?” Call on students to take you through the process of
plotting 5/8 (or some other non-unit fraction the students have not plotted yet).
Begin with a number line that has endpoints 0 and 1 and be sure to indicate that
the interval 0-1 is the whole. Have one student partition the interval into equal
parts using tick marks. Ask a student how/she knew to partition the whole into
that many equal parts. Have another student label the tick marks. Ask the student
to plot the fraction on the number line.
Accommodations

Special Needs: Peer remediation will be an easily obtained goal in this lesson.
Working with varying abilities has them make sure they have placed the
endpoints in the right location on a number line.

Advanced: Reinforce the mathematical language while they are doing the
assignment. For example, can they explain what a whole number or a fraction is?
Can they explain what a number line is or why it is important?

ELL: Monitoring how they use the mathematical language will help lead to
understanding. If they cannot verbalize what a tick mark is, make sure they are
able to show how to draw one and place it on a number line.
References
Fractions, Grade 3, Teacher Created Resources, 2011
Common Core Mathematics-Practice at 3 Levels; Levy, Jill; 2012
Blooms Taxonomy Questioning
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Fraction Bars
Purpose of the Lesson: Students will recognize and understand what an equivalent
fraction is. What it looks like (i.e., ½ would be half of colored circle) and why is it
important. Example, if you had two people eat a cookie what would they both get to
have the same amount? It is important because when you are looking at fractions, and
when the students continue on in the upper grades (fourth, fifth, etc.) the students will
learn how to simplify fractions, understanding equivalent fractions will help with this.
Also, at these stages of development students are very concerned about what is “fair.”
Using the example that by breaking something (Candy bars are good for representing
halves, cookies are good for representing fourths) into half and dividing it between two
friends means each person is given the equal amount of the product. To help support this
lesson, student will make their own fraction bar set. Fraction bars are another length
model. It is vital students gain experience with different types of fraction models so they
develop a more complete foundational understanding of fractions.
Common Core Standards: 3 NF 3 Explain equivalence of fractions in special cases, and
compare fractions by reasoning about their size.
a. Understand and generate simple equivalent fractions, e.g. 1/2=2/4, 4/6=2/3).
Explain why the fractions are equivalent, e.g., by using a visual fraction model.
b. Recognize and generate simple equivalent fractions, e.g., ½=/4, 4/6=2/3. Explain
why the fractions are equivalent, e.g., by using a visual fraction model.
Instructional Objective: By the end of the lesson, students will know how to recognize
and understand what an equivalent fraction is and why it is important to them in their
everyday lives. They will manipulate fraction bars to show equivalent fractions.
Assessment


Formal: Ticket out the door. Formal curriculum assessment if applicable
Informal: Support worksheet; in-class questioning
Materials

Fraction Bars. It is strongly recommend a teacher purchase a professional set from
a teaching store. The students can be giving a pre-made set that is copied onto
construction paper that students can cut out and make for use during this lesson,
or any other lessons that are applicable. Make sure the student set is a pre-labeled
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


fraction bar worksheet. It is strongly recommend the parts of the wheel are
copied onto construction paper. If it is run on white, the teacher can decide to
have students color code it to match the set being used to teach them.
Paper and Pencil
Supporting worksheet
Ticket out the Door
Procedures


Anticipatory Objective 20 mins: “Today we are going to learn about equivalent
fractions. Equivalent fractions are fractions that are equal – they are same
amount.” Have the students make their fraction bar with the provided template
the teacher has copied on construction paper. This will be the time when the
teacher can decide to color code the different fractions. Having an extra copy is
also recommended so if the teacher chooses to have the students cut out the
colored set, they can have the blank set to use as a work mat. Once this set is
complete, give students a minute just to play with their fraction bars. Maybe ask
the students a few simple questions, “can you show me ½?” The teacher could
also ask them to show their seatmate or partner what that looks like. Then ask the
students to see if they can figure out what ¼ will look like. It should be ok to fail
here; the student will be able to self-check because the bars will not fit properly if
they are not using the correct fraction bar. This is also a good time for a teacher
to informally look around to see who may have some background understanding
of some simple fractions or how the fraction bars may be used. During this time,
the teacher should be walking around to try and get an idea of who understands
how to “play” or line their fraction bar set so it represents the fraction correctly.
If the students have done it correctly all the bars will fit into a nice rectangle or
square, depending on the model used.
Main Learning Activity 40 mins: Demonstrate for the students how one whole
can break down to ½, then down to ¼, then down to 1/8. You may need to do the
same thing again; however, the second time they should do it with you using their
own manipulatives. After stopping the students from “playing” with their
fractions bar, ask them how they got to the ½ and ¼. How did they manipulate
the sections to fit properly? Is there a way for them to check their work? Could
this be done by breaking it down by thirds? What would that look like? Finally,
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lead them in a discussion that starts by saying, “How can we create a fraction the
same length as ½ but only using fourths?” This feedback will tell the teacher if
they are ready to attempt independent work.
It is advised the teacher complete one or two additional problems through
whole class instruction so they have an example. The teacher can choose to do
this right off the support worksheet. An example of this would be taking ½ and
breaking it down to sixths. They could represent this with their fraction bar, and
see that ½ is = to 3/6. The same example can be used with ¼, broken down into
two 1/8 bars, or pieces. ¼=2/8. The teacher can also use non-unit fraction
examples to help the students understand other ways equivalent fractions can be
used. For example 2/3 is equivalent to 4/6. Ask students to show 2/3 using their
fraction bars. Then ask how they could show an equivalent fraction using only
sixths. “How many sixths is equivalent to 2/3?” Wait for the students to use their
fraction bars to find the answer. “So the equivalent fraction is 4/6. How do you
know?” Teachers should listen for explanations that refer to the fact that 2/3
aligned end to end have the same length as 4/6 aligned end to end.
It is also important for students to have experience with equivalent
fractions that begin with smaller parts and are shown to be equivalent to fractions
with larger parts. For example 4/10 = 2/5. Ask students to show 4/10 using their
fraction bars. Then ask how they could show an equivalent fraction using only
fifths. “How many fifths is equivalent to 4/10?” Wait for the students to use their
fraction bars to find the answer. “So, the equivalent fraction is 2/5. How do you
know?” Teachers should listen for the same types of explanations as the previous
example except with tenths and fifths. As partners work, the students could find
fractions equivalent to ¾. The teacher would not give the students the hint to only
use eighths. It is important to remind the students they should only use fractions
that are all the same size.
Closure Activity 10 mins: Have students review the fractions taught. Then push
them to 1/3 and 1/5. Have the students identify and generate equivalent fractions
up to sixths, then repeat for fifths and tenths. Even if they do not get it right,
when it is taught it would not be a new thought process for the students and it will
give them one last chance to “play with their math toys.” Another important
opportunity would be to ask some closure questions on how and why these
concepts are important to their everyday lives. “What are some ways you could
reference fraction bars out in the real world?”
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Accommodations:



Special Needs: Using the fractions manipulate bar set. If the teacher has enough,
it would make sense to allow these students to use the professional manipulatives
so they are completely correct when trying to get every piece to fit together.
Advanced: Let them work together with like-minded students and have them
figure out exactly what 1/3, 1/5, etc. looks like versus just “playing” with the
fraction bars.
ELL: Have the students talk out exactly what they are doing, for example: “I
take the ½ bars and place them first.” You can have ELL students start with
steps, for example: “First, I…; Next I…; then I…”
References
Fractions, Grade 3, Teacher Created Resources, 2011
Common Core Mathematics-Practice at 3 Levels; Levy, Jill; 2012
Blooms Taxonomy Questioning
Example for Ticket out the Door
Name: ______________________Date:________________________
Draw a picture of what fraction bars look like broken down to ¼, then to 2/8 to show how
they are equivalent. Then explain how you know the two fractions are equivalent.
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Expressing One Whole/One as a Fraction
Purpose of the Lesson: The purpose of this lesson is to have the students be able to
recognize fractions that are equivalent to one whole or one. The students have already
learned about unit fractions (e.g., 1/3, 1/6, 1/8) and other fractions (e.g., ¾, 2/5, 5/8).
This lesson is an extension of those so students can understand that if there are four parts
shaded in a whole that is partitioned into fourths, then the entire whole is shaded. In
addition, students will equate one whole with the number 1.
Common Core Standards: 3C. Express whole[s] …fractions, and recognize fractions
that are equivalent to [a] whole. Locate 4/4 and 1 at the same point of a number line
diagram.
Instructional Objective: By the end of this lesson students will be able to identify
fractions that equal a whole and plot them on a number line. Example: 3/3=1.
Assessment


Formal: Formal assessment provided by the curriculum or one the teacher has
chosen.
Informal: Ticket out the door and the supplemental worksheet.
Materials



White boards, pens and erasers
Ticket out the Door
Pre-cut shapes if desired
Procedures

Anticipatory Set 15 mins: A common mistake with fractions is forgetting what
the numerator and denominator actually represents or tells the student working
through fraction math problems. Take a moment to explain that the numerator
tells us how much is shaded or being considered and the denominator tells us how
many equal parts make up the whole. Example: “If I had a circle that I cut into
thirds and I shaded two of the three equal parts, then 2/3 of the circle is shaded.”
Using a similar example, draw a circle on the overhead. Ask “If I had one whole
pie (1), not cut, and a whole pie cut into fourths (4/4) would they still represent
one whole pie?” Take the students through the process of identifying the
denominator for the first uncut pie (1) and then identifying the numerator (1) for
57
the same pie. Repeat the process for the second pie: identifying the denominator
(4) and identifying the numerator (4).” Then if I ate the 1 whole pies, and the one
cut into fourths, would I still have eaten two whole pies?”
1 whole
4/4


Main Learning Activities: Using squares and rectangles, continue on with the
same line of questioning. Have the students draw a square on their white boards.
Have them draw one whole (1) and divide the next square into thirds (3/3=1).
Ask one of them to explain what that looks like. You should hear something
about how regardless if the shapes is divided into three equal parts, if all the
pieces are colored it is still showing a whole. After the students have drawn the
undivided whole and the whole divided into three equal parts, make sure the
students label them as fractions (e.g., 1 or 1/1 and 3/3) and have the students hold
up their white boards for a quick assessment. Then, have each student develop
their own questions, using a shape of their choice and allow them to work in pairs
to demonstrate the whole concept.
In the second part of this lesson, the students will need to work on placing
fractions that represent a whole onto a number line. The first example could be
3/3 = 1. On the number line, have the tic marks already placed, then through
guided instruction have the students label the tick marks. Once they have
competed this example have them do a harder example, maybe 6/6=1. Make sure
to check in with them and ask them how they came to place the tick marks where
they did and why it is important to know that the tick marks have been placed in
the proper place. The students can accomplish this by also labeling all the tick
marks.
Make sure to check in with them and ask them how they came to place the tick
marks where they did and why it is important to know that the tick marks have
been placed in the proper place. The students can accomplish this by also
labeling all of the tick marks.
Closure Activity 15 mins: After the students have worked in their partner
groups, have them share out some of the examples they used with their partner.
Make sure they explain how and why they came to any of the conclusions they
did.
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Accommodations:


Special Needs/ELL: Give them pre-cut shapes to work with. If you’re going to
work with circles, give them circles, give them squares if you are going to work
with squares. Depending on where your ELL students are, you want to label the
shapes.
Advanced: Have the students explain how 4/4 is equal to 6/6; they will compare
two fractions equal to one whole.
References:
Blooms Taxonomy Questioning
Example for Ticket out the Door:
Name:___________________________Date_________________________________
I have two pies. One that is whole and the one that is partitioned into eight equal parts.
Write the fractions that represent each of the pies. Are they equivalent? How do you
know?
Plot 4/4
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Template for shapes if needed.
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Supplemental Work Sheet for Number Lines.
1. Plot 0 and 1
2. Plot ½
3. Place tick marks that show 3/3
4. Place tick marks that show 4/4
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5. Place tick marks for 6/6
6. Place tick marks showing 8/8
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Money, Money, Money
Purpose of the Lesson: The purpose of this lesson is for the students to be able to know
what ¼, ½, ¾, and one whole of a dollar represents. By the end of the lesson, the
students should be able to tell you parts of a dollar in money terms and fraction terms.
Example, ¼ of a dollar is a quarter.
Common Core Standards: 3.NF 3 Explain the equivalence of fractions in special cases,
and compare fractions by reasoning about their size.
e. Know and understand that 25 cents is a ¼ of a dollar, 50 cents is ½ of a dollar and 75
cents is ¾ of a dollar.
Instructional Objective: The students will be able to identify what coins equal 1/4, ½,
and ¾ of a dollar.
Example: 1 quarter equals ¼ of a dollar. Two quarters or .50 equals ½ of a dollar.
Assessment


Formal: Ticket out the Door for immediate assessment; formal curriculum
assessment if applicable
Informal: Support worksheet
Materials



Money. Four quarters, 8 dimes and 4 nickels. Try to have real coins, but fake
coins, or even printed out paper coins will work
support worksheet
Ticket out the Door
Procedures

Anticipatory Set: Start this lesson with a review of the coins. Talk about how
much a quarter is worth. The different combinations you can make to get to a
quarter. Go over what a dollar looks like and the different combinations for that.
You may need to do smaller quantities, it will depend on the class, however, good
for your ELL learners and any special needs students. Have a discussion with the
class on the importance of money with the class. “How is money used? When at
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the store or purchasing something why is it important to know how you are
handling your money? When you receive change in coins, what are some easy
ways to count it? If there is change given, in the amount of a dollar, how do you
count that? Would that amount equal ½ of a dollar? ¼ of a dollar? What would
happen if you were at the store and did not know how to count money?” Make it
relatable to their lives.


Main Learning Activities: After handing out basic supplies, four quarters, 8
dimes, and 4 nickels, have the students manipulate the coins in different ways to
make a dollar. How many dimes make a dollar? How many quarters make a
dollar? Next, concentrating on the quarters, have the students line them up.
Facilitate a class discussion that could look something like this:
“When we count quarters, what do we count by? Let‘s count the quarters to see
how much we have? Have many quarters make a dollar? If four quarters make a
dollar, then what fraction of a dollar is one quarter? How do you know? What
fraction of a dollar is two quarters? How do you know? What fraction of a dollar
is three quarters? How do you know? What fraction of a dollar is four quarters?
How do you know?”
What is another way that we could ¾ of a dollar using out coins? The teacher can
have the students do this activity with their seat partner, by the teacher.
Closure Objective: Using this opportunity to loop this lesson back to the
beginning, refer back to why it is important to understand the different parts of the
dollar and how it could affect their lives if they did not know how to count the
money. It is always entertaining to listen to what the students think they can
purchase for ¼ of a dollar.

Accommodations



Special Needs: I would make it a point to provide these students with real money
Advanced: Have these students use the dimes and the nickels to make 1/4, ½, and
¾.
ELL: Using the white board, have them draw the amount of money that equals
1/4, ½, and ¾ of a dollar. If the teacher does not have student white boards, the
teacher could make these on a computer prior to the lesson.
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References
Fractions, Grade 3, Teacher Created Resources, 2011
Common Core Mathematics-Practice at 3 Levels; Levy, Jill; 2012
Blooms Taxonomy Questioning
Ticket out the Door
Name:_______________________Date__________________________Fraction
Money Lesson
1. How much is ¼ of a dollar?
2. How much is ½ of a dollar?
3. How much is ¾ of a dollar
4. How many quarters equal a whole dollar?
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Support Worksheet for Money
Name:_______________________________________Date:______________________
_______
Using quarters, draw the following:
1. ½ of a dollar. This equals _____________cents
2. ¼ of a dollar. This equals______________cents
3. 1/5 of a dollar. This equals_____________cents
4. One whole dollar. This equals $____________
Using dimes and nickels draw the following:
1. ½ of a dollar. This equals _____________cents
2. ¼ of a dollar. This equals______________cents
CHALLENGE!
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3. 3/4 of a dollar. This equals_____________cents
4. One whole dollar. This equals $____________
Bonus Question
Ryan goes to the store to buy a candy bar for $0.75. He gets $0.25 for change. What
fraction of a dollar does he get back?
________________________________________________
Alexis goes to the store and spend $0.50 on a back of chips. What fraction of a
dollar does she have left to spend?
_______________________________________________________
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FRACTIONS OF SHAPES
Purpose of the Lesson: The purpose of this lesson is for students to be able to partition
geometric shapes into parts with equal areas so as to form unit fractions of the whole.
This is an area model of fractions meant to enrich the students’ foundational
understanding of fractions. Students will be exposed to the idea that equal fractions of
the same whole must have equal areas. The focus, however, will be on unit fractions.
Common Core Standards: 3.G Reason with shapes and their attributes
2. Partition shapes into parts with equal areas. Express the area of each part as a unit
fraction of the whole. For example, partition a shape into 4 parts with equal area and
describe the area of each part as ¼ of the area of the shape.
Instructional Objective: At the end of the lesson, students will be able to create unit
fractions by partitioning basic shapes, such as a square, triangle, circle, and rectangle into
parts with equal areas.
Assessment


Informal: Have students hold up their equally partitioned shapes. Supporting
worksheet
Formal: Ticket out the Door; Formal assessment if applicable
Materials




Light colored scrap paper
Writing tools
White Boards, pens and erasers
Pattern Blocks
Procedures

Anticipatory Set 10 mins: Hold up a rectangle piece of paper. Start by asking,
“What type of shape is this piece of paper? (Rectangle/quadrilateral). “How do
you know?” “Draw a line on this paper so that the paper shows two equal parts.
Each part of the paper is the same size. The parts are equal. How is this paper
divided? Each half of the rectangle is called ½.” Keeping in mind that a
rectangle can be split in half in different ways (e.g., vertically down the middle,
horizontally down the middle, then across the diagonal) ask the questions “How
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
do you know you have two equal parts? How do you know you have two equal
pieces? What fraction of the rectangle have you created? How do you know?”
Reinforce that each equal part of the rectangle is called ½.
Main Learning Activities 30 mins: Once the students have all held up their
rectangles and answered the questions, the teacher can assess the students’
understanding visually and verbally to determine if they are ready to move on. If
ready, challenge students to draw another line on the same piece of paper, or the
same rectangle, and break it into fourths or quarters. Continuing with the pattern
of questioning started in the anticipatory set, “how…” “Why…”etc., this will help
the teacher check for understanding. Because the students may have created
halves in different ways, there are many possibilities for how the students will
create fourths. Example of questions for the students: “Did you create equal
parts?” “How do you know this?” “How many equal parts did you create?” “What
do you call these equal parts?” “Can you explain why these equal parts are
called?” Also, have the students draw a circle and draw a line showing how to
divide it up into halves and fourths. Challenge the students to divide the other
circle into thirds. Since this can be a difficult task, get ideas from the students on
how to draw the lines so the circle is divided into three equal parts or thirds.
Depending on how the teacher is feeling her class is doing, the teacher could
challenge them to go into sixths and eighths. It could be recommended that the
teacher use the circle or rectangle. Another variation of this lesson would be to
either have the student draw, or given them predrawn squares and triangles have
them partition these shapes as well. The students by this point should have already
worked with a fraction wheel and the fraction bars.
Closure Activity 10 mins: Referring back to the “How did you get____?” and
“Why is called______?” questions that were used at the start of the lesson would
be a good wrap-around activity. The teacher could also have the students; before
the students turn in their work, ask the questions and have the students explain
what they did to find specific answers while referring to their worksheet.
Accommodations


Special Needs: Have them work with a partner. Have them use the pattern blocks
as much as possible.
Advanced: Pair them up with other advanced students and have them work
together to push partitioning their shapes to sixths and eighths. Or, pair them up
with students in need of remediation and have them reteach the concept. They
70

could also be challenged to try and partition more difficult shapes such as,
trapezoids, stars, or kites.
ELL: Check in with the students and have them use the correct vocabulary for
unit fractions and also “equal part.” Have them identify the shapes being worked
with and make sure they can create the proper fraction to it and refer to it by
name. Example: “this is ½ of a square (circle, rectangle, etc.), because…”
References
Fractions, Grade 3, Teacher Created Resources, 2011
Common Core Mathematics-Practice at 3 Levels; Levy, Jill; 2012
Blooms Taxonomy Questioning
Ticket out the Door
Name:____________________________________Date:__________________
How would you partition each of these shapes into halves, thirds, fourths, or
quarters?
Drawings of shapes if needed.
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REFERENCES
Ball, D., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching. Journal of
the American Federation of Teachers, Fall, 14-46.
California Department of Education (CDE). (2013). All curriculum frameworks.
Retrieved from http://www.cde.ca.gov/ci/cr/cf/allfwks.asp
California Department of Education (CDE). (n.d.). Common core state standards.
Retrieved from http://www.cde.ca.gov/re/cc/ccssfaqs2010.asp
Carson, L. (2000). Why supplemental curriculum helps teachers. Ezine@rticles.
Retrieved from http://ezinearticles.com
Clearinghouse, W. W. (2009). Best practices for RtI: Differentiaied reading instruction
for all students. Reading Rockets, 1-3.
Clearinghouse, W. W.(2010, September). Devloping effective fractions instruction for
kindergarten through 8th grade. Retrieved from
http://ies.ed.gov/ncee/wwc/pdf/practice_guides/fractions_pg_093010.pdf
Fennell, F. (2011). We need elementary mathematics specialist now, more than ever: A
historical perspective and call to action. NCSM Journal, 53-60.
Fensterwald, J. (2012, July 17). Two years plus and counting: Teachers prep for common
core. EduSource.
Gresham, G., & Little, M. (2012, August). RTL in math class, teaching children
mathematics. Teaching Children Mathematics, 19(1), 20-29.
92
Herbert, M. (2011, March 1). Are you ready to implement common core standards?
Retrieved from http://www.districtadministration.com/article/are-you-readyimplement-common-core-standards
Hightower, A. (2011). Improving student learning by supporting quality teaching: Key
issues, effective strategies. Bethesda, MD: Editorial Projects in Education, Inc.
Jennings, J. (2012). National standards. American School Board Journal, 46-47.
Jennings, J. (2012). Reflections on a half-century of school reform: Why have we fallen
short and where do we go from here? Retrieved from http://www.cepdc.org/displayDocument.cfm?DocumentID=392
Levy, J. (2012). Common core mathematics. Pelham, NY: Newmark Learning.
Lewin, T. (2010, June 21). Many school adopt national standards for their schools. The
New York Times, p. 3.
Mcmillan/McGraw Hill. (2008). Third grade math – 2008 edition – Teacher’s edition.
New York: Author.
Melton, B. (2011). Supporting implementation of the common core state stadards for
mathematics. Raleigh: North Carolina State University.
Mooney, N. J., & Mausach, A. T. (2008). Align the design: A blueprint for school
improvement. Alexandria, VA: ASCD.
The National Governors Association Center for Best Practices and Council of Chief State
School Officers (NGA & CCSSO). (2010). Common Core State Standards for
English language arts. Retrieved from www. corestandards.org/the-standards
93
Pattison, C., & Berkas, N. (2000). Critical issue: Integrating standards into the
curriculum. Retrieved from North Central Mathematics andScience Consortium at
North Central Regional Educationa Laboratory:
http://ncrel.org/sdrs/areas/issues/content/currliclum/cu300.htm
Powers, K., & Gamble, B. (2012). Authentic assessment. Retrieved from
http://www.education.com/print/authentic-assessment
Reeves, D. B. (2010, May 12). Common standards: From what to how common-core
standards should influence testing. Education Week, pp. 32-33.
Ross, P. (2010). Emerging issues in culrriculum and instruction. EDUC 897, 1-20.
Retrieved from Mason.gmu.edu/~pross1/portfolio/index.htm
Sawchuk, S. (2010). Putting new standards into practice a tough job. Education Week, 16.
U.S. Department of Education (DOE). (2010). Developing effective fractions for
kindergarten through 8th grade. Retrieved from
http://ies.ed.gov/ncee/wwc/pdf/practice_guides/fractions_pg_093010.pdf
Wu, H.-H. (Fall, 2011). Phonix rising. American Educator, 3-13.
Wurman, Z., & Wilson, S. (2012). The common core math standards: Are they a step
forward or back? Educationnext, 12, 3.