DYS math - Collaborative for Educational Services

DYS math - Collaborative for Educational Services
MATH
Teaching in DYS Schools
An Instructional Guide for Educators in the
Massachusetts Department of Youth Services
October 2006
•
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Executive Office of Health and Human Services
Department of Youth Services
27 Wormwood Street, Suite 400
Boston, MA 02210-1613
MITT ROMNEY
GOVERNOR
617.727.7575
FAX#: 617.951.2409
KERRY HEALEY
LIEUTENANT GOVERNOR
Timothy Murphy
SECRETARY
JANE E. TEWKSBURY, Esq.
COMMISSIONER
Dear Colleagues,
The Massachusetts Department of Youth Services and the Commonwealth Corporation are working
together to develop a comprehensive education and training system for the thousands of young
people placed in DYS custody every year by the juvenile courts. We are expanding and enriching
the continuum of options and opportunities—including high-quality education and training,
comprehensive case management, mentoring programs, and other services—to give incarcerated
youth in Massachusetts the knowledge, skills, and confidence they need to build a better future.
As a DYS teacher, you work closely with young people in custody, and you play a primary role in
helping them improve their lives. Many of your students have experienced failure in traditional
education systems, and trauma in other areas of their lives. Many demonstrate gaps or deficiencies in
learning that have prevented them from achieving academic success. By attending to students’
individual learning needs, you may be offering them the first effective educational experiences in
their lives.
The teaching professionals in DYS work hard to deliver high-quality, content-rich learning
opportunities that address the needs of all students. To help you adapt traditional math curricula and
textbooks for use in your classrooms, we have compiled vital information, stimulating resources,
culturally competent strategies, and sound pedagogical practices for teaching mathematics in
detention, assessment, and treatment facilities across the state.
We hope that this guide will assist you in your important work. The lessons, mini-units, and daily
problems in this guide have been modeled, tested and adapted in real classrooms—not just in theory.
This instructional guide is rooted in the rich experiences of math teachers in DYS and other facilities
serving vulnerable and at-risk learners, as well as the principles, strands and standards of the
Massachusetts Department of Education’s Mathematics Curriculum Framework.
Thank you for the hope and dedication that you bring to the young people in our care.
Sincerely,
Jane E. Tewksbury
Commissioner
TABLE OF CONTENTS
INTRODUCTION
Who are our students? What are their backgrounds and needs? What challenges
do DYS teachers face, and how does DYS organize to meet those challenges?
Reviewing philosophy, principles, and mission provides a birds-eye view of
educational programming for youth in DYS custody.
1
MATH AND CULTURE
15
FRAMING CURRICULUM AND INSTRUCTION
31
STRANDS AND EMPHASIZED STANDARDS
43
Connecting math to our students’ lives helps answer the perennial question,
“Why do I have to learn math? When will I ever use it?” Fundamental information,
examples, and resources about making mathematics instruction culturally
responsive to the young people we serve.
Defining the terms—curriculum, instruction, framework, strands, standards,
and assessments—helps teachers organize and plan differentiated instruction
that responds to students’ backgrounds, interests, and prior knowledge.
Given the extraordinary diversity and mobility in DYS classrooms, how can teachers
align their instruction with the Massachusetts Curriculum Framework?
Balancing consistency with flexibility is key. Teachers address all components of
a unified mathematics education by following a consistent calendar across all settings,
and teaching short mini-units and Problems of the Day that emphasize the most crucial
standards in each of the curriculum strands.
CURRICULUM UNITS, LESSONS, AND RESOURCES
Easily referenced information about Essential Questions, Sequencing, Background Resources
for teachers, Curriculum Resources for instruction, Problems of the Day, Sample Mini-Units,
Connecting Math to our students’ lives, and Pulling it all together with MCAS release items.
Data Analysis, Statistics, and Probability
September through November pages 63-95
Geometry and Measurement
April through June
Patterns, Relations, and Algebra
Number Sense and Operations
ASSESSMENT
December through March
Integrated in all strands
63
pages 97-131
pages 133-165
Definitions, guidance, and resources for balanced assessment using Bloom’s Taxonomy,
student work, rubrics, and other techniques to screen, diagnose, and measure students’
progress and attainment of learning standards.
173
INTRODUCTION
INTRODUCTION
This Mathematics Instructional Guide is the second in a series of five instructional guides prepared
by the Commonwealth Corporation for DYS teachers. The guides focus on major content areas in
DYS—English Language Arts, Math, Science, Life Skills, and Social Studies. These instructional
guides are aligned with the extensive program of professional development, training, and coaching
provided through a subcontract with the Hampshire Education Collaborative. All of the DYS
Instructional Guides share the same general outline, instructions for use, and alignment with both the
Massachusetts Curriculum Frameworks and the goals and principles of the DYS education system.
DYS EDUCATION PROGRAMS
3
PRINCIPLES for CURRICULUM & INSTRUCTION in DYS
4
MISSION and DYS EDUCATIONAL PHILOSOPHY
6
DYS STUDENT POPULATION
7
STATISTICAL PORTRAIT of DYS YOUTH
8
DYS STUDENTS WHO ARE ENGLISH LANGUAGE LEARNERS
DYS STUDENTS WITH SPECIAL NEEDS
1
10
11
INTRODUCTION
DYS EDUCATION PROGRAMS
Every day, the Department of Youth Services provides educational services to more than 1,500 young people in
58 sites across Massachusetts. In addition, DYS operates 38 day programs to serve youth transitioning back into
the community and residing with parents, guardians, or in independent living programs.
All DYS education and services focus on preparing youth to re-integrate successfully into their communities and
make successful transitions to public schools, alternative education programs, GED preparation, post-secondary
education, job skills training, or employment. Programs operate under contract with DYS, and are run by numerous
vendors and community-based organizations. The 58 DYS facilities in Massachusetts include:
Detention sites
for youth in the pre-commitment stage
Assessment sites
for youth committed to DYS and awaiting determination of placement
Treatment sites
short-term and long-term secure treatment programs for young people
DYS facilities across the state are united by shared principles, guidelines, professional development,
curricular materials, and coaching. Educational programming operates on a 12-month school year, with
a minimum of 27.5 hours of instructional services per week. DYS educational services strive to meet all
Massachusetts education standards, policies and procedures, including requirements for time and learning
and highly-qualified educator certification.
DYS education programs include:
Academic services, GED preparation, vocational education, life skills
programming, and/or post-secondary education services;
Educational liaisons who link across programs and with local school districts;
Special education services, provided through the Massachusetts Department
of Education’s Educational Services in Institutional Settings (ESIS);
Title I supplemental services, provided through federal entitlement funds;
Vocational/work programs including extended day, job training, and employment,
provided through partnerships with vocational-technical high schools and
WIA (Workforce Investment Act) youth programs.
The transient nature of the DYS student body, as well as students’ diverse ages, varied academic skills,
and the large numbers of students with special needs, pose unique challenges and opportunities in all
DYS educational programs.
MATHteaching in DYS schools
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3
INTRODUCTION
PRINCIPLES for CURRICULUM
Principles for quality curriculum and instruction in all subject areas reflects opportunities and challenges facing
throughout DYS. The principles are based on “Elements of Quality Instruction,” synthesized by Shirley Gilfether
AND INSTRUCTION in DYS SETTINGS
teachers and students in the DYS educational system. These eight principles guide quality teaching and learning
of the Hampshire Educational Collaborative (HEC), from research related to instruction for at-risk students.
CURRICULUM, INSTRUCTION, AND ASSESSMENT SHOULD BE TIED TO STATE FRAMEWORKS
YOUTH LEARN BEST BY SEEING CONNECTIONS ACROSS TOPICS
ALL YOUTH CAN LEARN
LITERACY IS THE BASE FOR ALL LEARNING
DYS youth are held to the same high learning standards as all other Massachusetts
youth. Our classrooms should reflect those standards and expectations. Rather
than measuring students against one another, we need to measure individual students
against the benchmark of learning standards.
A
Students(and adults) learn better when they can see the big picture and make
connections to their own experiences and to other things they know and have
learned in their lives. This means linking learning to essential questions, big
ideas, concepts, or themes.
Maintaining high expectations and establishing a “can do” attitude in the classroom
is essential to student success. Because teachers’ expectations have an enormous
affect on student achievement, one of the most powerful factors in student
achievement is the belief—on the parts of both teachers and students—that all
students can succeed.
In most classes, and throughout the rest of a student’s life, reading and
“decoding” information is the primary gateway to information, ways of
thinking, and new knowledge, skills, and abilities. Strong literacy skills—
including literacy in math—are essential for success in school and adult life.
GOOD QUESTIONING IS CRITICAL TO QUALITY LEARNING
LL YOUTH ARE DIFFERENT AS LEARNERS
Questions are a primary “tool of the trade” in education. There are different
types of questions that promote different types or levels of learning. (Bloom’s
Taxonomy, included later in this manual, provides a framework for different
levels of questions.) All students can work on questions, as long as they are
simple or complex, concrete or abstract at different levels. We need to include
different levels and types of questions in our teaching and assessment.
Students come to our programs with a variety of learning styles, intelligences,
cultural and educational backgrounds, and learning strengths. Teachers need to use
these differences, as well as students’ diverse interests, background experiences, and
prior knowledge to adjust curriculum and instruction to address learners’ needs and
increase their interest and engagement with the information they study and learn.
OUR YOUTH NEED ENCOURAGEMENT, PRAISE AND MOTIVATION TO LEARN
Many of our students come with a history of failure, low expectations, and criticism
in traditional schools. We need to build on what they can do and reinforce all
positive growth. By building on students’ interests and enthusiasm, reinforcing their
efforts and recognizing their growth, we can impact student learning even in a short
period of time.
Think about a learning experience you had as a teenager that was especially
memorable and powerful. Were any of these principles at work in that experience? Which one?
OUR YOUTH LEARN BEST WHEN ACTIVELY ENGAGED.
Like many youth, our students learn best when they are actively engaged and able
to make connections to their own experiences and real-world contexts. Many are
kinesthetic or visual-spatial learners, and most are looking for reasons why they
need to learn something. Teachers should strive to be “the guide on the side,” not
“the sage on the stage.” Teachers need to treat students as active learners, not as
passive recipients of instruction.
MATHteaching in DYS schools
Do you agree with all eight principles? Why? Why not?
Which principle is the biggest stretch for you in your classroom?
What are some ways you could incorporate that principle into your teaching?
4
5
INTRODUCTION
MISSION
The mission of the DYS Education program is to provide a comprehensive educational system that addresses the needs,
experiences, and goals of our youth. Through collaboration with local schools, community-based organizations, families,
and other resources, the DYS education system provides individualized education plans and services that focus on
literacy and numeracy skills, education and employment opportunities, and transitions to the community and workforce.
THE DYS STUDENT POPULATION
The DYS population is demographically diverse by race, ethnicity, language,culture, age, and economics. Our
students are educationally diverse with respect to their background knowledge, interests, aspirations, learning
styles, multiple intelligences, social-emotional strengths and challenges, and personal histories.
As teachers, we need to be sensitive to all the components of diversity, and use them as
strengths and opportunities to reach our students. When educators are sensitive to issues
related to diversity, we are better able to foster environments where differences are valued as
useful tools for teaching, learning, and engaging all students.
DYS EDUCATIONAL PHILOSOPHY
DYS is committed to providing an education program for all students in the DYS system that is in compliance with the
Massachusetts State curriculum standards. The curriculum guides that frame the instruction and assessment are organized
around key themes and essential learning outcomes that are modified for the various student placements (detention, assessment, and treatment). All education is delivered with an understanding of the diversity of the student population,
all curriculum, instruction and assessment planning includes components of differentiation, respect for cultural diversity,
and a commitment to enhance students’ overall literacy skills.
Topics, examples, and resources must be relevant to our students’ lives and experiences—not
culturally biased in a way that reflects an idealized white middle class experience.
When compared with the rest of the state’s populations, the young people in DYS custody
reflect disproportionately high percentages of youth of color (African-American and Latino),
youth for whom English is a second language, and students with learning disabilities. Topics,
examples,
and resources that are relevant to their lives and experiences—not culturally biased to reflect
an idealized white middle class experience—are especially crucial.
Some of the underlying conditions and commitments of the DYS educational system include:
of preparing a very diverse population for multiple pathways
A commitment to
that is aligned with state curriculum
frameworks and focused on curriculum standards for the most critical learning areas
For many students, classes in DYS settings offer an invaluable chance to re-engage with
learning. Some of our youth have done well in school, are proficient in mathematics, and
will use our classes to build and expand their success as learners. Others have not done well
in school, and many are significantly behind their peers in math. These students often are
discouraged and tend to avoid anything that resembles “school work.”
A fundamental recognition that our students come from a range of cultural and economic
that are often very different than those of their teachers
We have unique opportunities in DYS programs. By differentiating instruction to respond to
student’s readiness, interests, backgrounds and learning styles, we have the chance to:
A commitment to effective curriculum and instruction built around real-life situations that
are
to the diverse youth in our programs
Acknowledgement that many of our students have
that may or may not
have been recognized and attended to in their previous educational experiences
Tie instruction to students’ own experiences and background knowledge
Acknowledgement that the nature of detention, assessment, and treatment of youth in DYS
custody contributes to extremely high levels of
Provide our strongest learners with new, exciting experiences
A commitment to providing all teachers with common strategies and tools for curriculum,
instruction, and assessment, including concrete lessons that are referenced to larger learning
themes and
for various sites and student needs
Offer struggling learners successful experiences—often for the first time in their lives!
A commitment to providing teachers with high-quality materials, training, professional
development, references, and other resources that
each of the above on an
ongoing basis.
MATHteaching in DYS schools
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7
INTRODUCTION
STATISTICAL PORTRAIT
OF DYS YOUTH
The DYS system serves approximately 9,000 students per year, with roughly 1,500 students enrolled at a given time The DYS committed caseload decreased 9% between 1996-2006 DYS students in the detained population are 42%
white, 26% African American, 22% Latino, 3% Asian, and 7% other Youth are between 10 and 19 years old, with an average age of 15 years, 11 months at initial commitment On average, students are approximately 17 years old Young
people are placed in DYS custody for as little as one day to as long as several years Approximately 10% are indicted as Youthful Offenders and committed to DYS until the age of 21 Many DYS students have had a history of delinquency
prior to placement in DYS; Nearly half of DYS committed youth were placed in out-of-home residential placement by another agency prior to commitment to DYS Nearly 75% of DYS committed youth were adjudicated delinquent (found
guilty of a crime) and placed on probation at least once prior to their commitment to DYS Approximately 45% of the DYS committed population has been identified as qualified for special education services A large majority are between
2 and 4 years below grade level 20% of committed DYS students take psychotropic medications In only one of seven cases were the juvenile’s biological parents married and living together at the time of their child’s commitment to DYS
Fewer than 50% of the biological mothers and fathers of DYS committed youth had completed the 12th grade More than half of the committed population have received services from the Department of Social Services prior to commitment
Approximately 75% of committed girls have had DSS involvement The DYS system serves approximately 9,000 students per year, with roughly 1,500 students enrolled at a given time The DYS committed caseload decreased 9%
between 1996-2006 DYS students in the detained population are 42% white, 26% African American, 22% Latino, 3% Asian, and 7% other Youth are between 10 and 19 years old, with an average age of 15 years, 11 months at initial
commitment On average, students are approximately 17 years old Young people are placed in DYS custody for as little as one day to as long as several years Approximately 10% are indicted as Youthful Offenders and committed to
DYS until the age of 21 Many DYS students have had a history of delinquency prior to placement in DYS; Nearly half of DYS committed youth were placed in out-of-home residential placement by another agency prior to commitment to
DYS Nearly 75% of DYS committed youth were adjudicated delinquent (found guilty of a crime) and placed on probation at least once prior to their commitment to DYS Approximately 45% of the DYS committed population has been
identified as qualified for special education services A large majority are between 2 and 4 years below grade level 20% of committed DYS students take psychotropic medications In only one of seven cases were the juvenile’s biological
parents married and living together at the time of their child’s commitment to DYS Fewer than 50% of the biological mothers and fathers of DYS committed youth had completed the 12th grade More than half of the committed population
have received services from the Department of Social Services prior to commitment Approximately 75% of committed girls have had DSS involvement The DYS system serves approximately 9,000 students per year, with roughly 1,500
students enrolled at a given time The DYS committed caseload decreased 9% between 1996-2006 DYS students in the detained population are 42% white, 26% African American, 22% Latino, 3% Asian, and 7% other Youth are between
10 and 19 years old, with an average age of 15 years, 11 months at initial commitment On average, students are approximately 17 years old Young people are placed in DYS custody for as little as one day to as long as several years Approximately 10% are indicted as Youthful Offenders and committed to DYS until the age of 21 Many DYS students have had a history of delinquency prior to placement in DYS; Nearly half of DYS committed youth were placed in outof-home residential placement by another agency prior to commitment to DYS Nearly 75% of DYS committed youth were adjudicated delinquent (found guilty of a crime) and placed on probation at least once prior to their commitment to
DYS Approximately 45% of the DYS committed population has been identified as qualified for special education services A large majority are between 2 and 4 years below grade level 20% of committed DYS students take psychotropic
medications In only one of seven cases were the juvenile’s biological parents married and living together at the time of their child’s commitment to DYS Fewer than 50% of the biological mothers and fathers of DYS committed youth
had completed the 12th grade More than half of the committed population have received services from the Department of Social Services prior to commitment Approximately 75% of committed girls have had DSS involvement The
DYS system serves approximately 9,000 students per year, with roughly 1,500 students enrolled at a given time The DYS committed caseload decreased 9% between 1996-2006 DYS students in the detained population are 42% white,
26% African American, 22% Latino, 3% Asian, and 7% other Youth are between 10 and 19 years old, with an average age of 15 years, 11 months at initial commitment On average, students are approximately 17 years old Young people
are placed in DYS custody for as little as one day to as long as several years Approximately 10% are indicted as Youthful Offenders and committed to DYS until the age of 21 Many DYS students have had a history of delinquency prior
to placement in DYS; Nearly half of DYS committed youth were placed in out-of-home residential placement by another agency prior to commitment to DYS Nearly 75% of DYS committed youth were adjudicated delinquent (found guilty
of a crime) and placed on probation at least once prior to their commitment to DYS Approximately 45% of the DYS committed population has been identified as qualified for special education services A large majority are between 2 and
4 years below grade level 20% of committed DYS students take psychotropic medications In only one of seven cases were the juvenile’s biological parents married and living together at the time of their child’s commitment to DYS Fewer than 50% of the biological mothers and fathers of DYS committed youth had completed the 12th grade More than half of the committed population have received services from the Department of Social Services prior to commitment
Approximately 75% of committed girls have had DSS involvement The DYS system serves approximately 9,000 students per year, with roughly 1,500 students enrolled at a given time The DYS committed caseload decreased 9%
between 1996-2006 DYS students in the detained population are 42% white, 26% African American, 22% Latino, 3% Asian, and 7% other Youth are between 10 and 19 years old, with an average age of 15 years, 11 months at initial
commitment On average, students are approximately 17 years old Young people are placed in DYS custody for as little as one day to as long as several years Approximately 10% are indicted as Youthful Offenders and committed to
DYS until the age of 21 Many DYS students have had a history of delinquency prior to placement in DYS; Nearly half of DYS committed youth were placed in out-of-home residential placement by another agency prior to commitment to
DYS Nearly 75% of DYS committed youth were adjudicated delinquent (found guilty of a crime) and placed on probation at least once prior to their commitment to DYS Approximately 45% of the DYS committed population has been
identified as qualified for special education services A large majority are between 2 and 4 years below grade level 20% of committed DYS students take psychotropic medications In only one of seven cases were the juvenile’s biological
parents married and living together at the time of their child’s commitment to DYS Fewer than 50% of the biological mothers and fathers of DYS committed youth had completed the 12th grade More than half of the committed population
have received services from the Department of Social Services prior to commitment Approximately 75% of committed girls have had DSS involvement The DYS system serves approximately 9,000 students per year, with roughly 1,500
students enrolled at a given time The DYS committed caseload decreased 9% between 1996-2006 DYS students in the detained population are 42% white, 26% African American, 22% Latino, 3% Asian, and 7% other Youth are between
10 and 19 years old, with an average age of 15 years, 11 months at initial commitment On average, students are approximately 17 years old Young people are placed in DYS custody for as little as one day to as long as several years Approximately 10% are indicted as Youthful Offenders and committed to DYS until the age of 21 Many DYS students have had a history of delinquency prior to placement in DYS; Nearly half of DYS committed youth were placed in outof-home residential placement by another agency prior to commitment to DYS Nearly 75% of DYS committed youth were adjudicated delinquent (found guilty of a crime) and placed on probation at least once prior to their commitment to
MATHteaching in DYS schools
8
9
INTRODUCTION
DYS STUDENTS WHO ARE ENGLISH LANGUAGE LEARNERS
DYS STUDENTS WITH SPECIAL NEEDS
Unfamiliarity with certain words, idiomatic expressions, and cultural references from popular
Approximately 45% of DYS students have documented special education
needs. Many experts believe that the percentage of youth with special needs
in DYS is actually much higher. Furthermore, even students whose special
needs have been recognized, and for whom Individualized Education Plans
(IEPs) have been developed, may arrive in DYS facilities and classrooms
long before their papers do. With all this in mind, it is clear that all teachers
in DYS must understand and play important roles in meeting the needs of
special education students.
http://www.doe.mass.edu/sped
Unfamiliarity with rules and cues of English and US mathematics (for example, switching
SPECIAL EDUCATION RIGHTS AND PROCESSES:
Nationally, English Language Learners are the fastest growing segment of the school-age population. Not surprisingly, a growing number of students in the DYS system are English Language Learners (ELL). Mathematics instruction,
like all content instruction, must be tailored/customized to the learning needs of linguistically and culturally diverse
students.A diverse group, with varied levels of English fluency and academic backgrounds in other languages,
English Learners in DYS face a number of special challenges in mathematics, including:
US fables, nursery rhymes, or particular media;
periods and commas for decimal places or place indicators, fewer linguistic markers for
singular and plural nouns);
Fear of looking foolish in front of peers because of limited skills in English.
HELPING ENGLISH LANGUAGE LEARNERS LEARN MATH CONTENT:
http://www.tesol.org/s_tesol/index.asp
DYS teachers may wish to refer to the TESOL standards
for content instruction, which have been used by many
states in writing of state standards and for developing and implementing math curriculum. The web address for
TESOL (Teachers of English to Speakers of Other Languages) is provided above. Additionally, the following tips
and ideas may also help DYS teachers differentiate instruction for ELL students:
Allow for clarification in students’ native language, if possible;
Emphasize new mathematical terms and present them in context;
Have English Learners work in small groups with other students;
Help students become aware of their own learning processes;
Make connections between students’ prior experience and new material;
Make sure that the material taught, and the language used to teach it,
is appropriate for your students’ levels of English fluency;
Model all activities first, to enable students to observe what is expected;
Present each concept in a number of different ways;
Use a variety of question types, including those that promote
higher-level thinking skills;
Use graphic displays (pictures, charts, labels) and real-world
objects to help students understand the material being taught.
MATHteaching in DYS schools
Selected
strategies
from
Making
Content
Comprehensible for
English
Language
Learners:
The SIOP
Model,
Second
Edition.
Allyn &
Bacon,
2003, and A
Framework
for
Teaching
English
Learners,”
WestEd
Regional
Educational
Laboratory,
2005.
10
In Massachusetts, the special education system is based on the federal
special education law—the Individuals with Disabilities Education Act
(IDEA)—in combination with the state’s special education law
(MGL c. 71B). These laws outline rights and responsibilities to ensure
that students receive a free and appropriate public education in the “least
restrictive” environment that meets their learning needs, and require that special needs students receive an education
that is designed to meet their unique needs through an Individualized Education Plan (IEP).If a student is assessed as
having special needs, an IEP team should be formed to write the student’s plan. Once the team has
Sources
completed the IEP and it has been approved by the parent, guardian, or surrogate parent, it is
include
Guide to
implemented by the student’s school, reviewed by the IEP team at least once a year, and revised if
the
necessary. An IEP establishes annual goals for the student, broken down into short-term objectives or
Individualbenchmarks. The goals may address academic, social or behavioral needs, relate to physical needs, or
ized Education
address other educational needs. Some students have what is known as a “504 plan;” this is a plan for
Program,”
students who have disabilities that do not interfere with their ability to progress in general education.
Office of
Special
Education
and
Rehabilitative
Services,
U.S. Dept
of Education, July,
2000. and
A Parent’s
Guide to
Special
Education,
The
Federation
for
Children
with
Special
Needs
and the
Mass
Dept of
Education
11
THE ROLE OF THE REGULAR EDUCATION TEACHER:
• Work with the education liaison to obtain a copy of each student’s IEP, and modify your
instruction to meet the goals stated in the IEP;
• Seek out professional development, coaching, and training to help you understand the
special needs of your students;
• Coordinate your teaching with the Special Education instructor;
• If a student does not have an IEP and you feel that he or she has special needs, work
with your school director and the parent or guardian to begin the evaluation process;
• Participate in the IEP team and use your knowledge of the student’s progress and needs
to contribute to developing or revising the student’s IEP;
• Support the student with good teaching strategies that are differentiated for all learners;
• Be an advocate for all of your students to ensure that they receive appropriate support.
INTRODUCTION
MATHteaching in DYS schools
12
MATH AND CULTURE
WHAT IS CULTURALLY RESPONSIVE TEACHING?
and WHY DOES IT MATTER?
16
MAKING OUR CLASSES MORE CULTURALLY RESPONSIVE
18
EXAMPLES OF CULTURALLY RESPONSIVE CURRICULA,
CLASSES, and PROGRAMS
20
WHERE TO LEARN MORE
24
15
MATHEMATICS AND CULTURE
WHAT IS CULTURALLY RESPONSIVE TEACHING? WHY DOES IT MATTER?
Diversity does not refer only to ethnicity or race.
Differences in social class, family culture, geographic
and religious backgrounds, and learning styles are all
reflected in our classrooms as important components
of diversity.
DYS teachers and students often come from very
different cultural, social, and economic backgrounds.
In addition to the strong commitment and empathy
that teachers have for their students, we must also
offer them culturally responsive instruction. But what
does cultural responsiveness mean—on the ground,
and in the classroom?
In 1994, Gloria LadsonBillings
defined
Culturally Responsive
Teaching as “a pedagogy
that recognizes the
importance of including
students’
cultural
references in all aspects
of learning.” Teaching
math in culturally
responsive ways means
using students’ own
habits, experiences, and
cultural references to connect to real-world
experiences with numbers, shapes, patterns, chance,
and measurement.
Teaching is most successful when we connect new
skills with students’ prior knowledge. When we work
with students’ own experiences, we help them
recognize and appreciate the role of mathematical
knowledge in their own lives and activities.
To motivate our students to become mathematically
literate, and to prepare students to use math in school,
careers, and daily life, we must engage our students in
mathematics. The essential question to ask, and to encourage our students to ask, is, “What are the problems that my community is facing, and how can I use
math to understand and help solve them?” (This question is borrowed from the math and social justice
website: RadicalMath.org)
Think about a teacher in your past who understood and respected you. What did that teacher do that
made you feel that you were truly respected?
Think about a classroom setting in which you did not feel understood or respected. What were the
factors that made you feel that way? How did this affect your learning experiences in that setting?
Cultural competence lies in finding
examples, models, problems, and
illustrations that draw not on
teachers’ cultural experiences, but
on the cultural references of students.
All of our students are adolescents with rich histories
of problem solving, reasoning, and communicating.
Think about ways that your students already use
mathematical thinking skills like Comparing,
Classifying, Generalizing, Predicting,
Quantifying, and Searching for
Patterns.
MATHteaching in DYS schools
Remember a classroom activity or experience that related to your own life, approach or perspective.
What did your teacher do to create this opportunity? Was your learning different?
Consider keeping a personal journal about teaching. Dr. Sonia Nieto’s book, The Light in Their
Eyes: Creating Multicultural Learning Communities (Teachers College Press, 1999) offers many
examples of how keeping a journal expanded teachers’ sense of themselves in a diverse world.
Mathematics lessons and activities that are grounded
in the real issues in students’ lives will increase their
knowledge and application of problem-solving and
reasoning skills. Culturally responsive teaching is the
key to increasing student engagement in the classroom, and answering the perennial question,
”Why do I have to learn math?
When will I ever use it?”
16
17
MATHEMATICS AND CULTURE
MAKING OUR CLASSES MORE CULTURALLY RESPONSIVE
To be effective in developing the academic skills of our students, DYS schools must create culturally responsive
learning environments that honor the diversity of our students.
When we build learning communities that are culturally responsive, we
tremendous possibilities for both teachers and students to experience
, and
.
The work of creating these environments is not easy, but it is worth exerting the effort and
enduring the mistakes we make along the way.
Think about a particular skill or piece of information that you learned earlier
in your life and is still with you.
Every teacher and student brings a different set of strengths and experiences to the endeavor of
creating classrooms that really work. Nonetheless, there are a number of common features that
all culturally responsive curricula and classroom practices share.
What did a teacher, parent, religious leader, or family friend do to help
build a bridge from what you already knew to new skills or information?
Are there ways you can help your students build bridges between their
knowledge and cultural backgrounds and new math skills? What are
some of the ways?
Think about the terms on the left-hand page—
, and
. Do these concepts
resonate in your home, family, or community life?
Start from where the students are
Are aware of their own influence on the culture of the classroom
Create places of learning for both students and teachers
What does “ownership” mean in the context of DYS teaching? Do you
think “connection” and “transformation” are reasonable goals to strive
for in one’s work life?
Recognize and are critical of cultural biases and stereotypes
Connect to students’ real lives and experiences
Promote active and dynamic learning
Pose engaging questions that stimulate investigation and discovery
Respect the different rates and ways that people learn
Encourage different perspectives and approaches
Ensure that instruction is academically rigorous for all students
Express your recognition that each person in the room—students, teachers, and line staff—
has a story to share, and that each person’s story is continually unfolding and being revised.
No one is static, no one is a closed book or a finished chapter.
The expression, “We are all works in progress” is especially meaningful in DYS settings.
MATHteaching in DYS schools
18
19
MATHEMATICS AND CULTURE
EXAMPLES OF CULTURALLY RESPONSIVE CURRICULA, CLASSES, AND PROGRAMS
EXAMPLES OF CULTURALLY RESPONSIVE CURRICULA, CLASSES, AND PROGRAMS
Although the use of mathematics is universal, math is not
culturally neutral. Successful methods for learning, calculating, memorizing and communicating about math
actually differ quite a lot across cultures.
Culturally Situated Design Tools allow students and
teachers to explore mathematics with depth and care,
using cultural artifacts from specific times, places,
and cultures. The Culturally Situated Design Tools
(CSDT) website provides free standards-based
lessons and interactive “applets” that help students
and teachers explore the mathematics and knowledge
systems embedded in such cultural artifacts as:
MATH IS A LANGUAGE…AND THERE ARE MANY DIALECTS
Mary Jane Schmitt, author of The Answer is Still the
Same: It Doesn’t Matter How You Got It! (Peppercorn
Books and Press, 1991), learned from immigrants in her
adult education classes that there are significant differences in the methods that people around the world learn
to compute. To honor the background and prior knowledge of diverse students, and to help teachers understand
the variety of math procedures around the world, Schmitt
provides numerous examples and explanations of the
ways that notation and computation differ by country,
culture, and historic patterns of colonization.
Many of these differences are also explored through the
“Algorithm Collection Project,” an initiative based at the
California State University in Sacramento that systematically collects algorithms for addition, subtraction,
multiplication, and division, and publishes them on the
web at http://www.csus.edu/indiv/o/oreyd/ACP.htm_
files/Alg.htm.
Educators in many countries are surprised to learn that
day-to-day algorithms for basic arithmetic operations
vary by culture and by national origin. Given
the many elements of diversity in DYS, it’s
important to recognize that many of our
students come to us with strengths and background knowledge that are as valuable to
them as they are unfamiliar to us.
Imagine, for a moment, having to re-learn an
algorithm that you were taught as a child.
Would you continue to compute in the familiar way? Would you try to disguise your
methods? Would you “forget” what you had
previously learned?
Different ways to write numbers
from The Answer Is Still the Same:
It Doesn’t Matter How You Got It!
In some countries (including the US) a decimal point—
—is used to separate a whole number from a
decimal fraction. Therefore:
1,000 means one thousand;
3.5 means 3 units and 5 tenths;
Decimal equivalents of fractions may be written
either as .6 or 0.6, for example.
In many other countries (including Puerto Rico,
England, and most of Europe), the comma—
—is used instead of the decimal point, and
vice-versa. Therefore:
1.000 means one thousand;
3,5 means 3 units and 5 tenths;
The zero is always written in the whole number
place, for example, 0,6.
Different ways to compute long division
These different ways of writing numbers (above) vary in
small but recognizable ways. Diverse methods for computing multiplication and long division, on the other
hand, vary much more significantly. The samples below,
from The Algorithm Collection Project, are just three
examples among the many ways that people around the
world would compute the same division problem.
3,285714
Youth subculture graffiti (teaching
Cartesian and polar coordinates)
Native American beadwork
(teaching Cartesian coordinates)
Cornrow curves (teaching Fractal geometry)
Percussion rhythms (teaching Ratios & fractions)
The simulation software and teaching materials are copyrighted
to Ron Eglash and Rensselaer Polytechnic Institute, and are
generously provided on the web by Dr. Ron Eglash, a professor
at Rensselaer Polytechnic Institute and the author of African
Fractals: Modern Computing and Indigenous Design (Rutgers
University Press, 1999). Some examples of CSDT lessons are
also included in the Curriculum Resources section of this guide;
see “Connecting Math to Our Students’ Lives” for each strand of
mathematics instruction.
Each CSDT topic comprises a number of resources that enable
teachers to integrate the topics into standards-based math
instruction. Resources for each topic include:
A section on cultural background and history
3.285714
3,285714
A tutorial on the math topic and its connection to
cultural artifacts and systems of knowledge
Software (applets) that enable teachers and students to simulate the
development of these artifacts, and
Links to extensive teaching materials—including lesson
plans, pre- and post-tests, and samples of student work
from a wide variety instructional settings.
Look at the division problems to the right.
Could you compute long division in a way that is
unfamiliar to you? How do you think you would
reconcile the “new” way with the old?
MATHteaching in DYS schools
CULTURALLY SITUATED DESIGN TOOLS: Teaching Math through Culture
20
21
http://www.rpi.edu/~eglash/csdt.html
MATHEMATICS AND CULTURE
EXAMPLES OF CULTURALLY RESPONSIVE CURRICULA, CLASSES, AND PROGRAMS
RADICAL MATH: SOCIAL JUSTICE IN THE MATH CLASSROOM
Many teachers find that teaching math from a social
justice perspective is a powerful way to address state
and national standards while preparing students for
standardized tests, math-based careers, and mathrelated college majors. To assist teachers in locating
appropriate materials, the Radical Math website
organizes curricula and other teaching resources that
are purposefully relevant to students’
lives and experiences.
Launched in 2006 by a public high
school teacher in Brooklyn, New
York, the website encourages both
teachers and students to ask, “What
are the problems that my community
is facing, and how can I use math to
understand and help solve them?”
SUMMARY AND REFLECTION
When teaching a particular mathematics strand (i.e.,
Number Sense, Data, Algebra, or Geometry), teachers
are likely to search for curriculum resources by math
topics. For example, a search for resources to teach
(listed as a math topic) locates a lesson
where students use real-life data sets to explore
whether race is a factor in mortgage loans.
“ What are the problems that my
community is facing, and how
can I use math to understand
and help solve them? ”
The Radical Math website help teachers respond to
this essential question with a user-friendly site that
contains free downloads and links to more than 700
lesson plans, articles, charts, graphs, data sets, and
additional websites. The site is especially easy to
search, and materials can be found by searching either
by math topic, social justice issue, or type of resource.
Culturally responsive instruction addresses the specific
interests, concerns, and experiences of students in the
classroom. Teachers who have a sense of what issues
motivate and are interesting to their students may want
to search the site by social justice issues.
For example, searching for resources related to
(listed as a
social justice issue) reveals numerous articles and
data sets, as well as a curriculum that uses algebraic
and statistical methods to explore differences between
the general population and the military.
Similarly, searching under
locates items relating gentrification to voter turnout,
Wal-Mart, and housing and rental costs. These
resources explore the issue of gentrification through
percents, percent growth, scatterplot graphing, lines
of best fit, correlation, and randomization.
MATHteaching in DYS schools
EXAMPLES OF CULTURALLY RESPONSIVE CURRICULA, CLASSES, AND PROGRAMS
DO
DON’T
approach this work with depth,
sensitivity, and care
connect math problems to cultural heritage
in trivial ways, for example, simply
changing the names in the word problems
examine mathematical practices in
their social contexts, showing how math
can reflect important knowledge
systems in different cultures
challenge cultural stereotypes and
genetic myths about math; these are
damaging to both minority and
majority group members
When seeking resources for teaching
,
teachers will find links to lessons on Polyrhythms,
drumming, and other musical applications of Algebra,
as well as the complex mathematics that are
associated with origami.
stereotype indigenous peoples as
historically isolated or alive only in the past
analyze indigenous designs
(e.g., Zuni beadwork or African village
architecture) from a western view only
Adapted from Dr. Ron Eglash’s remarks at the National Council of Teachers of Mathematics (NCTM)
2001 Annual Meeting
Another teacher, searching for engaging ways to
teach
, will find a
multidisciplinary unit that explores water use issues
through researching and charting the relative costs of
water usage and oil spills.
Resources referenced under
are
linked to a wide range of units, lessons, and data sets
including household income, wage negotiations, SAT
statistics, and casualties in Iraq.
Compare what you know about culturally responsive teaching to your understanding of
multicultural education. Do you think these are two descriptions of the same thing, or do the
approaches seem different? How are they similar? In what ways are they different?
These are just a few examples of lessons, articles, and
other resources indexed on the Radical Math website.
Some examples are included in the Curriculum
Resources section of this guide (see “Connecting
Math to Our Students’ Lives”), and all can be viewed
and downloaded from the Radical Math website.
Think about the guidance offered in the DO and DON’T boxes, above. How would you
express these considerations in tour own words?
Do you think the examples on the previous pages (from The Answer is Still the Same, the
Algorithm Collection Project, Culturally Situated Design Tools, and Radical Math) respond to
the guidance offered in the DO and DON’T boxes? How do they exemplify or violate this
guidance?
http://www.radicalmath.org
22
Think about other math instruction you have seen or read about that address students’ diverse
backgrounds. How do those lessons exemplify or violate this guidance?
23
MATHEMATICS AND CULTURE
WHERE TO LEARN MORE
RADICAL EQUATIONS: CIVIL RIGHTS FROM MISSISSIPPI TO THE ALGEBRA PROJECT
Robert Moses, a leader in the Civil Rights Movement in Mississippi, believes
that transforming math education is as urgent today as winning the right to vote
was in the Jim Crow South of the early 1960’s. In Radical Equations: Civil
Rights from Mississippi to the Algebra Project (Beacon Press, 2001), Moses and
Charles E. Cobb, Jr. make a powerful case that math literacy—algebra in particular—is a “gatekeeper” in the fundamental struggle for citizenship and equality.
Because higher-order thinking and problem-solving skills are necessary to enter
into the economic mainstream, youth without these skills will be tracked into
an economic underclass. “Economic access and full citizenship depend
crucially on math and science literacy,” they observe, yet “illiteracy in math is
acceptable the way illiteracy in reading and writing is unacceptable.”
After winning a MacArthur Foundation award, Robert Moses spent two
decades teaching and experimenting in middle school classrooms in Cambridge and Mississippi, and went on to establish the Algebra Project to address
the lack of economic access faced by children from communities of poor people and people of color. The Algebra Project, and the book that describes it,
offer powerful arguments and concrete examples for using culturally-responsive curricula to help poor youth and young people of color learn high-order
thinking and problem-solving skills that are crucial for success.
WE CAN’T TEACH WHAT WE DON’T KNOW: WHITE TEACHERS, MULTIRACIAL SCHOOLS
WHERE TO LEARN MORE
EDWARD DEJESUS AND THE MOVEMENT FOR “YOUTH CULTURAL COMPETENCE”
The Youth Development and Research Fund (YDRF) champions a
different kind of cultural competency. Edward DeJesus, YDRF’s
founder and a long-time youth worker, is passionate about connecting
with young people to produce educational gains, and implores adults to
become “Youth Culturally Competent.”
DeJesus warns that when educational and workforce development
programs marginalize and disrespect youth culture, they lose the
opportunity to validate the identities that students have taken on as a
necessary part of growing up, and to harness those identities and
activities to promote the importance of education and work.
Many adults are surprised to learn that youth culture encompasses a
great many positive voices and visions. DeJesus and the YDRF have
released two books, Countering the Urban Influence: Reclaiming the
Stolen Economic Fortunes of America’s Youth, and Makin’ It: The HipHop Guide to True Survival, as well as Strength of a Nation, a hip-hop
CD and curriculum guide about HIV/AIDS, drugs, violence, and
survival. The YDRF website and all of YDRF’s work demonstrates
both the wisdom and the ways to use positive peer pressure, youth
involvement, and youth popular culture to help youth make decisions
that promote life, freedom and future economic opportunities.
RETHINKING MATHEMATICS: TEACHING SOCIAL JUSTICE BY THE NUMBERS
Rethinking Mathematics offers practical examples of lessons and
source materials that promote rigorous, high-quality math instruction.
More than 30 articles and lesson plans use mathematics to explore
topics that resonate with students’ lives, including racial profiling,
mortgage approval rates, concentrations of liquor stores in poor
neighborhoods, inequity of resources in public schools, military
recruitment, and more.
Gary Howard’s We Can't Teach What We Don't Know: White
Teachers, Multicultural Schools (Teachers College Press, 1999)
addresses questions of diversity across all curriculum areas.
Considered a pioneering book in examining the role of white leaders
and classroom teachers in an increasingly multicultural society, it
forms the basis for a good deal of anti-racism work in the U.S. and
around the world.
Editors Eric Gutstein and Bob Peterson argue persuasively that lessons
like these increase students’ interest in learning math. When students
recognize math skills as powerful tools they can use to advocate for
themselves and their communities, they become increasingly motivated
to learn and pursue higher levels of mathematics.
Howards’s work as an author, educator, and trainer is based on his
own journey of personal and professional transformation. Exploring
and embracing one’s personal identity as a white person is a necessary
process, Howard believes, in answering the question, “How do I be
anti-racist without appearing anti-White?”
Like all materials produced by Rethinking Schools, Rethinking Mathematics is grounded in the belief that students’ home cultures and
languages are strengths on which teachers can build, not deficiencies
for which they must compensate. Rethinking Schools also maintains
a website at www.rethinkingschools.org/publication/math where
teachers can download primary sources and classroom materials that
support many of the lessons presented in this book.
We Can't Teach What We Don't Know is written in a manner that is full
of hope, personal history, and vision. Particularly for white educators
who have felt angry, insulted, confused or hurt by other anti-racist
literature, this is a book that can be comforting, eye-opening and
inspiring.
MATHteaching in DYS schools
http://www.ydrf.com
24
25
MATHEMATICS AND CULTURE
WHERE TO LEARN MORE
INTEGRATING ISSUES OF SOCIAL, POLITICAL, AND ECONOMIC JUSTICE INTO YOUR MATH CURRICULUM
MATH
JUSTICE ISSUE
Adding
Basic Family Budgets
Determine how much money a family needs to survive,
live comfortably, etc.
Averages
Mayan Mathematics
Learn how to add, subtract, multiply in a base 20 system
Fractions
Geometry
Graphing
www.epinet.org/content.cfm/datazone_
fambud_budget
www.dpsk12.org/programs/almaproject
/pdf/MayanMathematics.pdf
JUSTICE ISSUE
Logarithms
Growth Rates
People, prisoners, AIDS cases, health factors, etc.
Inequalities
Percents
Union Salaries
www.bls.gov/ces/home.htm#data
Union officials and management often have different ways
www.unionstats.com/
to come up with the “average salary” of a worker. Use real
data to understand how the mean, median, or mode could
each be used, and the difference each makes in the “average.”
Combinations The Lottery
How the Lottery works, why it’s nearly impossible to win,
and the economic damage it causes
Exponents
USEFUL WEBSITES
MATH
Compound Interest & Population Growth
Growth/decline of food and water resources, cities
War Budgets
Compare budgets for defense department to budgets for
other social services; compare how money spent on
military operations could be used for other services
www.warresisters.org/piechart.htm
http://costofwar.com/index.html
www.brainzip.com and
Liquor Stores
maps.google.com
Look at how many liquor stores are within a 1-mile radius
of a given location, then compare with other neighborhoods www.epa.gov/enviro/wme
Environmental Racism
Determine the density of toxic waste facilities,
factories, dumps, etc, in the neighborhood
www.census.gov
Gentrification
Change in density in a neighborhood, by race and income
Line Graphs
Incarceration rates for different populations
Scatterplot Graphs and/or Regression
Correlation between % of any two of the following factors:
percent of population that are people of color, rates of
poverty, pollution, crime, health issues (i.e., rates of
asthma, AIDS, diabetes, obesity)
MATHteaching in DYS schools
Probability
Proportions
Percent of each race in total population vs. incarcerated
vs. in the military, vs. killed in the war, vs. dropping out
of high school, vs.college graduates, etc.)
Racial Profiling
Explore the probability that the subject of a random
traffic stop would be a person of color
Rates & Slope Prison growth
Rates of different races and genders being incarcerated,
compared to growth of high school graduates, and
compared to growth of funding for higher education
Population growth
In different countries (also good for looking at a
population density and comparison with resources)
Resource Density
Density of banks compared to check-cashers and
pawn shops in rich vs. poor communities
Statistics
www.ojp.usdoj.gov/bjs/glance/tables/c
orr2tab.htm
www.census.gov and
www.infoshare.org
26
Interest & Compound Interest
Making money through a savings account, Increasing debt
on a credit card, payday loans, tax refund loans, mortgage
payments, how APR works, comparing different APR’s
Growth Rates
Growth in rates of homeless, poverty, people in jail, etc.
http://mathforum.org/library/drmath/vi
ew/56122.html
www.census.gov/ipc/www/worldpop.ht
ml
Small Business
Create algebraic inequalities that describe a business’s
limits(ie,. time, supplies), and graph multiple inequalities
to determine number of products (x,y) that maximize profit
Community Surveys
Teach students how to write surveys, then survey the
community about any issue. Statistical analyses (i.e.,
averages, ranges, frequency tables, graphing, correlation,
percents, testing hypotheses, variance, standard deviation)
can be used to understand the results
USEFUL WEBSITES
www.sbma.gov
http://globalatlas.who.int/
www.demos.org/page37.cfm
www.nedap.org/resources/documents/F
INALRALSREPORT.pdf
http://nedap.org/programs/fairlending.html
www.census.gov/hhes/www/poverty/his
tpov/histpovtb.html (poverty)
www.cdc.gov/nchs/data/hus/hus05.pdf
(health)
www.census.gov/hhes/www/housing.html
(housing)
www.racialprofilinganalysis.neu.edu/
and www.census.gov
www.ojp.usdoj.gov/bjs/abstract/p04.htm
http://nces.ed.gov/pubs2002/dropout91
_97/all_tables.asp
http://coe.ilstu.edu/grapevine/Welcome
.htm
www.nedap.org/programs/mapgallery.
html
www.datacenter.org and other
sources, above, provide relevant
statistical data for analysis
References and information compiled by Jonathan Osler, founder of the Radical Math website, and excerpted from
A Guide for Integrating Issues of Social, Political, and Economic Justice into Mathematics Curriculum. The guide is a work-inprogress, offered as a free download from www.radicalmath.org. Feedback is requested at [email protected]
27
MATHEMATICS AND CULTURE
FRAMING
CURRICULUM
AND INSTRUCTION
CURRICULUM and INSTRUCTION—DEFINING THE TERMS
33
A SYSTEM for CURRICULUM & INSTRUCTION in DYS
34
GUIDING PRINCIPLES: CURRICULUM & INSTRUCTION IN MATH
35
FROM FRAMEWORKS… TO INSTRUCTION
36
DIFFERENTIATION of INSTRUCTION
38
STRATEGIES for DIFFERENTIATION
31
39
FRAMING CURRICULUM & INSTRUCTION
CURRICULUM AND INSTRUCTION—DEFINING THE TERMS
All fields of endeavor have specialized vocabulary or jargon. This kind of terminology can be very useful, enabling
practitioners to use a kind of “insider shorthand” to communicate with others in the field. It is essential, however, that
terms be defined, so that the same words mean the same things to all who use and hear them. Some of the most
frequently used terms in the field of education include:
CURRICULUM
Ideas, skills, processes, and outlooks that educators identify as important for students
to learn in each content area; curriculum is the “what” of education
INSTRUCTION
Interaction between teacher and student, or the actual activities that communicate
and review knowledge, understanding, or skill; instruction is the “how” of education
FRAMEWORKS
Curriculum frameworks in each content area, consistent throughout Massachusetts
STRANDS
Major organizing principles for learning in each content area
STANDARDS
ASSESSMENTS
Learning goals in each content area, delineated within each strand
Various methods to gather evidence of students’ expanding knowledge and skills
Good curriculum translates broad, overarching frameworks, strands, and standards into concrete lessons, mini-units, ts,
Good curriculum translates broad, overarching frameworks, strands, and standards into concrete lessons, mini-units,
daily activities, assessments, and supporting materials. These provide the means through which teachers engage
their students, and lead them through actions that will result in students’ meeting their learning objectives.
In every classroom, teachers build their curricular programs by:
Defining specific goals for student learning, based on the
frameworks, strands and standards
Assessing current levels of knowledge and skill among
all students in the classroom
Planning activities and selecting materials that will support
those goals and are differentiated for different learners
Implementing activities and using materials that are appropriate to the
needs, interests, backgrounds, and experiences of their students.
Assessing students’ growth in meeting specific goals
In DYS settings, instruction needs to be especially interactive and engaging. The challenge is to help DYS teachers
stress academic rigor and simultaneously differentiate instruction to respond to differences in the backgrounds,
abilities, interests, and learning styles among diverse and highly mobile students.
33
FRAMING CURRICULUM & INSTRUCTION
A SYSTEM for CURRICULUM & INSTRUCTION in DYS
GUIDING PRINCIPLES: CURRICULUM & INSTRUCTION IN MATH
“ Math must speak to our
students’ desires to understand
their lives: who they are and
how the world functions. ”
LEARNING
DYS mathematics teachers face multiple challenges
planning and delivering effective curriculum. These
include addressing the Massachusetts Mathematics
Frameworks and MCAS, minimizing duplication of
content, teaching a transient population, and
addressing the need for differentiated instruction.
Youth come to DYS with vastly differing sets of
skills, abilities, and background
knowledge, as well as their own hopes,
fears, experiences and aspirations.
Additionally, there is a range of different
settings and educational programs
within the DYS education system, as
well as great variation in the length of
time young people spend in our
programs.
Math must speak to our students’ desires to understand
their lives: who they are and how the world functions.
Relevance and applicability are crucial for personal growth,
to motivate learning, and to bring meaning to what happens
in the classroom.
We must also be attentive to the wide
range of possibilities for our students
when they leave the DYS system. Our
most fundamental goal is to prepare
them for a successful future outside of
DYS, which may include returning to
high school, passing MCAS or GED
examinations, and entering the Job
Corps, employment, or a college,
university, or other learning option.
“ Many factors —both educational
and not—constantly impact our
students’ learning, and they all
have an effect on what we can
do in the DYS classroom.”
A great many factors —both educational
and not—constantly impact our students’ learning,
and they all have an effect on what we can do in the
DYS classroom. An equally intense challenge is
engaging students in the learning process. DYS
students are racially and ethnically diverse. Most are
racial minorities, many are poor, and the
overwhelming
majority
have
experienced
discrimination, inequity, trauma, and violence.
Given the extraordinary range of variables in the DYS
system, our challenge is to develop a system of education that
is coherent and consistent, as well as flexible. To meet these
challenges, we have developed a highly adaptable
curriculum, organized around broad mathematical topics and
reflecting key principles for math instruction in DYS settings.
The Massachusetts Mathematics Framework is built around five principles, or philosophical statements. These
principles should guide the construction and evaluation of mathematics curricula in the DYS educational system.
A fuller description of these guiding principles is provided in the Framework document. (All teachers in the DYS
system have, and should refer, to a copy of this extensive and important document.)
1
2
3
4
5
TEACHING
Technology is an essential tool in a mathematics education.
EQUITY
All students should have a high quality mathematics program.
ASSESSMENT
As teachers, we must strive to meet our students where
they are now, build learning activities around their interests,
and tailor instruction to address
their individual learning styles
and preferences. Reaching out in
these ways enables students to
use their own background knowledge to acquire and retain new
skills and new learning.
34
An effective mathematics program focuses on problem solving and requires teachers who have a
deep knowledge of mathematics as a discipline.
TECHNOLOGY
Assessment of student learning in mathematics should take many forms to inform instruction and
learning.
Look over the 5 Guiding Principles from the Massachusetts Framework.
“ Relevance and applicability are
crucial for personal growth,
to motivate learning, and to
bring meaning to what happens
in the classroom. ”
MATHteaching in DYS schools
Mathematical ideas should be explored in ways that stimulate curiosity, create enjoyment of
mathematics, and develop depth of understanding
Choose 3 that seem especially important for youth in DYS settings. How can you promote these
principles in your classroom?
Choose 1 that seems like the greatest stretch for a math class for DYS youth. Why does this seem to be
a stretch? Talk to your colleagues and coaches about how to promote that principle in your classroom.
Which of the principles would be most important to use in planning curriculum, instruction, and
assessment at the particular site where you teach? Why?
35
FRAMING CURRICULUM & INSTRUCTION
FROM FRAMEWORKS …TO INSTRUCTION
FROM FRAMEWORKS…TO CULTURALLY RESPONSIVE INSTRUCTION
MASSACHUSETTS CURRICULUM FRAMEWORK
MASSACHUSETTS CURRICULUM FRAMEWORK
Present important knowledge in each subject area, broken down
into key categories known as:
Present important knowledge in each subject area, broken down
into key categories known as:
STRANDS
STRANDS
Break each category down into specific, measurable
learning objectives known as:
Break each category down into specific, measurable
learning objectives known as:
STANDARDS
STANDARDS
Indicate what information teachers need to teach by using, researching,
downloading, or developing their own lesson plans, problems,
and units, known as:
Indicate what information teachers need to teach by using, researching,
downloading, or developing their own lesson plans, problems,
and units, known as:
CURRICULUM (or curricula, the plural)
CURRICULUM (or curricula, the plural)
Shapes interactions with students that build upon
students’ knowledge and skills through:
Shapes interactions with students that build upon
students’ knowledge and skills through:
Projects,
problem-solving
drawn from students’ own
MATHteaching in DYS schools
36
37
and providing materials
in different ways to address
students’ diverse learning
styles, backgrounds,
needs, challenges, and
res
strengths
pons
lly
backgrounds
Interacting
Cultura
scenarios, and examples
experiences and
INSTRUCTION
i ve ins t
t
c
ru
ion
C
y responsi
l
l
a
r
ve
ltu
riculum
cur
u
INSTRUCTION
FRAMING CURRICULUM & INSTRUCTION
DIFFERENTIATION OF INSTRUCTION
THE ACCESS CENTER
By matching strategies for instruction to students’’
characteristics, teachers can strengthen learning for
everyone in the classroom.
Differentiating instruction allows all students to
access the same classroom curriculum. The
curriculum itself is not changed.
WHAT CAN BE DIFFERENTIATED?
PROCESS
CONTENT
How a student accesses
the material
What material is being learned
STRATEGIES FOR DIFFERENTIATION
The Access Center in Washington, DC has created a
Math Differentiation Brief available for free download
from the internet. This brief overview helps teachers
implement differentiation strategies through:
Assessing cognitive readiness by, for
example, using KWL charts (charts that
ask students to identify what they already
Know, what they Want to know, and
what they have Learned about a topic),
pre-tests, or other assessments;
Inventorying student interests by including
students in planning processes; and
PRODUCT
Identifying students’ learning profiles by
determining their
How a student shows what has
been learned
WHAT SHOULD
DIFFERENTIATION RESPOND TO?
Differences in
COGNITIVE READINESS
Students’ skill levels and
background knowledge
MATHteaching in DYS schools
Differences in
INTERESTS
Topics that are motivating and
respond to students’ interests
Differences in
LEARNING PROFILES
Learning styles
Grouping Preferences
Environmental
Grouping Preferences (does the
student work best individually, with a
partner, or in a large group?), and
Environmental Preferences (does the
student need lots of space or a quiet
area to work?);
The Access Center’s Math Differentiation Brief
provides concrete examples of differentiation for
specific topics in mathematics, including measurement,
fractions, geometry, collecting and analyzing data,
making inferences, and algebraic concepts.
http://www.k8accesscenter.org
Preferences
38
Learning Styles (is the student a
visual, auditory, tactile, linguistic or
kinesthetic learner?)
39
THE ACCESS CENTER
While acknowledging that there is no one-size-fitsall recipe for differentiation, Dr. Carol Ann Tomlinson
presents a comprehensive description of specific
strategies for differentiation in her book, How to
Differentiate Instruction in Mixed-Ability Classrooms, published by the Association of Supervision
and Curriculum Development (2nd edition, 2001).
Examples of some differentiation strategies include:
STRATEGY
RESPONDS TO
Compacting
Readiness
Independent Study
Interest
Interest Centers
or Interest Groups
Interest & Readiness
Flexible Grouping
Interest, Readiness, &
Learning Profiles
Readiness & Learning
Profiles
Tiered Assignments
and Products
Multiple Levels
of Questions
Learning Contracts
Choice Boards
Readiness
Readiness & Learning
Profiles
Readiness, Interest, &
Learning Profiles
The Access Center has also provided a a very useful
chart for planning and implementing strategies for
differentiation. Like the Math Differentiation Brief ,
the chart is is available at no cost to download from
The Access Center’s website.
FRAMING CURRICULUM & INSTRUCTION
MASSACHUSETTS MATHEMATICS
CURRICULUM FRAMEWORK
STRANDS AND STANDARDS
OF THE MASSACHUSETTS CURRICULUM FRAMEWORK
45
FOUR STRANDS of the MATH CURRICULUM FRAMEWORK
WHICH STANDARDS ARE EMPHASIZED IN DYS?
SUMMARY OF MATH STRANDS AND EMPHASIZED STANDARDS
KEY ELEMENTS FOR DEVELOPING and PLANNING QUALITY INSTRUCTION
46
49
57
58
FROM STRANDS and STANDARDS TO DAILY INSTRUCTION
COMPONENTS (TEMPLATE) for a MINI-UNIT
WHAT CAN YOU DO IN A DAY? (PROBLEMS OF THE DAY)
60
CURRICULUM RESOURCES FOR THE FOUR MATH STRANDS
DATA ANALYSIS, STATISTICS, and PROBABILITY
Pages 63-95
GEOMETRY and MEASUREMENT
Pages 133-165
PATTERNS, RELATIONS, and ALGEBRA
NUMBER SENSE and OPERATIONS
43
Pages 197-131
Integrated into each strand; see pages 82-83, 118-119, and 150-151
Massachusetts Math Curriculum Framework
59
STRANDS & STANDARDS
63
FOUR STRANDS of the MATH CURRICULUM FRAMEWORK
In addition to the key DYS principles and guiding principles for math discussed in the previous section, the strands
and standards articulated in the Massachusetts Mathematics Framework are crucial components for planning
curriculum and instruction in DYS settings. The framework divides mathematics learning into four strands:
DATA ANALYSIS, STATISTICS, AND PROBABILITY
PATTERNS, RELATIONS, AND ALGEBRA
GEOMETRY AND MEASUREMENT
NUMBER SENSE AND OPERATIONS
Each of these four strands includes many detailed learning standards. Because DYS students’ mathematics competencies
span a wide age and grade range, this guide focuses on mathematics learning standards for grades 7-10. Analysis has
shown that certain math standards predominate in the MCAS for grades 8 and 10. These particular learning objectives are
considered key learning standards within DYS, because they occur with great frequency on the MCAS and are extremely
useful and applicable in employment, life skills,
and future learning. The strands and key
standards emphasized in this manual are
outlined on the following pages.
The full Framework … also
provides excellent curriculum
and instruction, as well as
suggestions and resources for
assessment.
All DYS teachers should also have and refer
to their own copies of the complete Framework
document. The full Framework not only offers
detailed grade-appropriate standards, but also
provides excellent curriculum, instruction, and
assessment suggestions and resources. A downloadable PDF of the entire Framework is available on the Massachusetts
Department of Education’s website.
A flexible curriculum
… emphasizes key standards
and integrates important
principles, high standards,
essential questions, and
big ideas.
45
As discussed in previous chapters, the challenges to
developing an organized and systematic curriculum for
the DYS educational system include high mobility as
well as extraordinary diversity of ages, skills,
background knowledge, personal backgrounds and
history, and more. In response to these challenges, the
Department of Youth Services, Commonwealth
Corporation, and the Hampshire Education
Collaborative worked together to develop a flexible
curriculum that emphasizes key standards, reflects
important principles, standards, essential questions,
and big ideas, and is aligned with an extensive
program of professional development and coaching.
Massachusetts Math Curriculum Framework
STRANDS & STANDARDS
WHICH STANDARDS ARE EMPHASIZED IN DYS?
In each of the four strands of mathematics, we emphasize certain key standards in DYS settings. Math teachers and
coaches have selected the “emphasized standards” outlined on the following pages because they meet the following
broad criteria:
1
2
3
4
5
6
These standards help identify
that can guide student learning
and help them to think about the larger picture and use math concepts in all aspects of their lives
They are tied to the principles for math instruction and to the strands and standards from the
Massachusetts Mathematics Framework; they promote
and rigor.
Questions and problems associated with these standards occur with the greatest
on the Massachusetts Comprehensive Assessment System exams.
These math concepts and materials are most
employment, life skills, and future learning.
and fundamental to
These ideas are broad enough to guide teachers in planning instruction throughout the year,
while allowing enough
for individual
teachers and programs to choose many of their own
materials and meet individual students’ needs.
Emphasized standards provide a measure of
between programs, and provide a mechanism for
sharing lessons and units among DYS teachers.
http://www.doe.mass.edu/frameworks/math/2000/final.pdf
The four strands of mathematics instruction, and all standards within
these four strands, are outlined in the full framework document
available on the web at the address listed above.
The following pages review those standards that are emphasized in DYS
instruction. These pages are color-coded for readers’ ease of reference.
47
Massachusetts Math Curriculum Framework
STRANDS & STANDARDS
DATA ANALYSIS, STATISTICS, and PROBABILITY
WHAT
Formulate questions that can be addressed with data, and
collect, organize, and display relevant data to answer them.
Select and use appropriate statistical methods to analyze data.
Develop and evaluate inferences and predictions that are
based on data. Understand and apply basic concepts of
probability.
WHEN
September through November
EMPHASIZED
STANDARDS
Students engage in problem solving, communicating,
reasoning, connecting, and representing as they:
8.D.3
Find, describe, and interpret appropriate measures of central tendency (mean, median,
and mode) and spread (range) that represent a set of data. Use these notions to compare
different sets of data.
8.D.2
8.D.4
10.D.1
10.D.2
49
Select, create, interpret, and utilize various tabular and graphical representations of data,
e.g., circle graphs, Venn diagrams, scatterplots, stem-and-leaf plots, box-and-whisker
plots, histograms, tables, and charts. Differentiate between continuous and discrete data
and ways to represent them.
Use tree diagrams, tables, organized lists, basic combinatorics (“fundamental counting
principle”), and area models to compute probabilities for simple compound events, e.g.,
multiple coin tosses or rolls of dice.
Select, create, and interpret an appropriate graphical representation (e.g., scatterplot,
table, stem-and-leaf plots, box-and-whisker plots, circle graph, line graph, and line
plot) for a set of data and use appropriate statistics (e.g., mean, median, range, and
mode) to communicate information about the data. Use these notions to compare
different sets of data.
Approximate a line of best fit (trend line) given a set of data (e.g., scatterplot). Use
technology when appropriate.
Massachusetts Math Curriculum Framework
STRANDS & STANDARDS
PATTERNS, RELATIONS, and ALGEBRA
WHAT
WHEN
EMPHASIZED
STANDARDS
8.P.1
8.P.2
8.P.4
8.P.5
8.P.6
8.P.7
8.P.9
8.P.10
10.P.1
10.P.2
10.P.6
10.P.7
51
Understand patterns, relations, and functions. Represent and analyze
mathematical situations and structures using algebraic symbols.
Use mathematical models to represent and understand quantitative
relationships. Analyze change in various contexts
December through March
Students engage in problem solving, communicating,
reasoning, connecting, and representing as they:
Extend, represent, analyze, and generalize a variety of patterns with tables,
graphs, words, and, when possible, symbolic expressions. Include arithmetic and geometric
progressions, e.g., compounding.
Evaluate simple algebraic expressions for given variable values, e.g., 3a2 – b for a = 3 and b = 7.
Create and use symbolic expressions and relate them to verbal, tabular, and graphical representations.
Identify the slope of a line as a measure of its steepness and as a constant rate of change from
its table of values, equation, or graph. Apply the concept of slope to the solution of problems.
Identify the roles of variables within an equation, e.g., y = mx + b, expressing y as a function of
x with parameters m and b.
Set up and solve linear equations and inequalities with one or two variables, using algebraic
methods, models, and/or graphs.
Use linear equations to model and analyze problems involving proportional relationships. Use
technology as appropriate.
Use tables and graphs to represent and compare linear growth patterns. In particular, compare
rates of change and x- and y-intercepts of different linear patterns.
Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative,
recursive (e.g., Fibonacci Numbers), linear, quadratic, and exponential functional relationships.
Demonstrate an understanding of the relationship between various representations of a line.
Determine a line’s slope and x- and y-intercepts from its graph or from a linear equation that
represents the line. Find a linear equation describing a line from a graph or a geometric
description of the line, e.g., by using the “point-slope” or “slope y-intercept” formulas. Explain
the significance of a positive, negative, zero, or undefined slope.
Solve equations and inequalities including those involving absolute value of linear expressions
(e.g., |x - 2| > 5) and apply to the solution of problems.
Solve everyday problems that can be modeled using linear, reciprocal, quadratic, or exponential
functions. Apply appropriate tabular, graphical, or symbolic methods to the solution. Include
compound interest, and direct and inverse variation problems. Use technology when appropriate.
Massachusetts Math Curriculum Framework
STRANDS & STANDARDS
GEOMETRY AND MEASUREMENT
WHAT
WHEN
EMPHASIZED
STANDARDS
8.G.1
8.G.2
8.G.3
8.G.4 and 10.G.5
8.M.3
8.M.4
10.G.1
10.G.2
10.G.4
10.G.6
10.M.1
10.M.2
Analyze characteristics and properties of 2- and 3-dimensional geometric
shape,s and develop mathematical arguments about geometric relationships. Specify locations and describe spatial relationships using coordinate
geometry and other representational systems. Apply transformations
and use symmetry to analyze mathematical situations. Use visualization,
spatial reasoning, and geometric modeling to solve problems. Understand measurable attributes of objects and the units, systems, and
processes of measurement. Apply appropriate techniques, tools, and
formulas to determine measurements
April through June
Students engage in problem solving, communicating, reasoning, connecting, and
representing as they:
Analyze, apply, and explain the relationship between the number of sides and the sums of the
interior and exterior angle measures of polygons.
Classify figures in terms of congruence and similarity, apply these relationships to solving problems.
Understand the relationships of angles formed by intersecting lines, including parallel lines cut
by a transversal.
Demonstrate an understanding of the Pythagorean theorem and apply the theorem to the
solution of problems. Solve simple triangle problems using the triangle angle sum property
and/or the Pythagorean theorem.
Understand concepts and apply formulas and procedures for determining measures, including
area and perimeter/circumference of parallelograms, trapezoids, and circles. Use formulas to
determine the surface area and volume of rectangular prisms, cylinders, and spheres. Use
technology as appropriate.
Use ratio and proportion (including scale factors) in the solution of problems, including
problems involving similar plane figures and indirect measurement.
Identify figures using properties of sides, angles, and diagonals and identify figures’ type(s) of symmetry.
Draw congruent and similar figures using a compass, straightedge, protractor, and other tools
such as computer software. Make conjectures about methods of construction, and justify
conjectures by logical arguments.
Apply congruence and similarity correspondences (e.g., ∆ABC ≅ ∆XYZ) and properties of the
figures to find missing parts of geometric figures, and provide logical justification.
Use the properties of special triangles (e.g., isosceles, equilateral, 30-60-90º, 45-45-90º) to solve problems.
Calculate perimeter, circumference, and area of common geometric figures such as trapezoids,
parallelograms, circles, and triangles.
Given the formula, find the lateral area, surface area, and volume of prisms, pyramids,
spheres, cylinders, and cones (e.g., find the volume of a sphere with a specified surface area).
NUMBER SENSE and OPERATIONS
WHAT
Understand numbers, ways of representing numbers, relationships among numbers, and
number systems. Understand meanings of operations and how they relate to one another.
Compute fluently and make reasonable estimates.
WHEN
This strand is fundamental to students’ mathematical understanding, but may be
tedious if taught alone.
EMPHASIZED
STANDARDS
8.N.1
8.N.3
8.N.8
8.N.10
8.N.12
10.N.1
10.N.2
10.N.3
10.N.4
55
Topics and resources to integrate the Number Sense and Operations strand throughout
the year are included within the chapters for each of the other strands in mathematics.
Compare, order, estimate, and translate among integers, fractions and mixed numbers
(i.e., rational numbers), decimals, and percents.
Use ratios and proportions in the solution of problems, in particular, problems involving
unit rates, scale factors, and rate of change.
Demonstrate an understanding of the properties of arithmetic operations on rational
numbers. Use the associative, commutative, and distributive properties; properties of
the identity and inverse elements (e.g., -7 + 7 = 0; 3/4 x 4/3 = 1); and the notion of
closure of a subset of the rational numbers under an operation (e.g., the set of odd
integers is closed under multiplication but not under addition).
Estimate and compute with fractions (including simplification of fractions), integers,
decimals, and percents (including those greater than 100 and less than 1).
Select and use appropriate operations—addition, subtraction, multiplication, division, and
positive integer exponents—to solve problems with rational numbers (including negatives).
Identify and use the properties of operations on real numbers, including the associative,
commutative, and distributive properties; the existence of the identity and inverse elements
for addition and multiplication; the existence of nth roots of positive real numbers for any
positive integer n; and the inverse relationship between taking the nth root of and the nth
power of a positive real number.
Simplify numerical expressions, including those involving positive integer exponents
or the absolute value, e.g., 3(24 – 1) = 45, 4|3 – 5| + 6 = 14; apply such simplifications
in the solution of problems.
Find the approximate value for solutions to problems involving square roots and cube
roots, without the use of a calculator.
Use estimation to judge the reasonableness of results of computations and of solutions
to problems involving real numbers.
MassachusettsMath
MathCurriculum
CurriculumFramework
FrameworkSTRANDS
STRANDS &
& STANDARDS
STANDARDS
Massachusetts
KEY ELEMENTS FOR DEVELOPING AND PLANNING QUALITY INSTRUCTION
These key elements for developing and planning for quality instruction are based on “Elements of Quality
Instruction,” synthesized by Shirley Gilfether of the Hampshire Educational Collaborative. For more information
and synthesis from this expansive body of research, please see “Principles for Curriculum and Instruction in DYS
Settings” on pages 10 and 11 of the Introduction to this manual.
CLEARLY IDENTIFY AND LINK YOUR LEARNING OBJECTIVES
Define what you want students to know, understand, and be able to do as a result of this curriculum and
instruction. Link your objectives to the emphasized standards in Massachusetts Curriculum Frameworks
PLAN YOUR CURRICULUM AND INSTRUCTION WITH THE END IN MIND
Engage in “backward planning;” before designing the mini-unit or lesson, decide how students will
demonstrate their knowledge and understanding through assessments, products, or performances.
KNOW YOUR LEARNERS
Before beginning, do a quick pre-assessment of students’ background knowledge on the topic. Preassessment may be difficult in DYS settings with especially high turnover, but questioning and KWL
techniques are quick and helpful. This will help you decide what needs to be taught explicitly, what
adjustments you will to make to differentiate instruction for individual learners, and how you may group
students for project work.
PLAN FOR DIVERSITY, CREATIVITY, AND REALISM
You should plan to use a variety of materials and resources so that the needs of students are met, students
are engaged, and learning is meaningful. Lesson activities should be designed to help students prepare for
the final assessment, product, or performance, as outlined in your learning objectives.
REFLECT AND ADJUST
After the unit or lesson is done, make quick notes of what went well, what didn’t go well, and what should
change. Continuous improvement is a hallmark of a successful and inspiring teacher.
CLEARLY COMMUNICATE THE LEARNING OBJECTIVES
Talk about the learning objectives
with your students. Post the objectives in your room and refer back to them often. Use them to frame
assessment.
CLEARLY COMMUNICATE CLASSROOM EXPECTATIONS
Similarly, classroom rules, procedures, expectations, and consequences should be clearly articulated,
posted, referenced, and enforced. When you are in control of your classroom, you can ensure that learning
takes place.
BEGIN WITH RITUALS AND ROUTINES
Start every lesson or every day with a ritual or routine. This may be an engaging hook to introduce the
lesson (for example, by reading an excerpt from a news report that relies on numbers and statistics), or it
might be a short activity that energizes students, sets the tone for the class, and incorporates a quick lesson.
57
Massachusetts Math Curriculum Framework
STRANDS & STANDARDS
COMPONENTS (TEMPLATE) FOR A MINI-UNIT
FROM STRANDS & EMPHASIZED STANDARDS TO DAILY INSTRUCTION
UNIT OBJECTIVES
To support the emphasized standards, DYS teachers use
mini-units, lessons, and Problems of the Day that are
directly tied to learning standards.
LEARNING STANDARDS
Standards-based planning involves having a clear
relationship among at least these seven things :
1
TOPICS or Essential Questions
3
ACTIVITIES and tasks to teach and demonstrate
2
STANDARDS to be taught and assessed
knowledge and skills in pursuit of learning objectives
4
PRODUCTS and performances that form the
5
CRITERIA for assessment, based on the standards
6
SCORING GUIDES to assess and communicate about student learning
7
PRE-ASSESSMENT
Based on what students should know, understand, be able to do
How will you determine students' readiness for this unit?
What data will you collect?
What survey of prerequisite learning will you use?
(i.e., KWL charts, journal prompts, oral surveys)
RESOURCES AND MATERIALS
basis for assessment of progress toward learning
objectives
OUTLINE OF LESSONS
and the learning objectives associated with them
Should reflect differences in students' readiness to learn (prerequisite
learning), interests (choices), and learning profiles (learning styles,
environmental and grouping preferences)
Lesson tasks and activities to support students' achievement of learning objectives
INTRODUCTORY
INSTRUCTIONAL
EXEMPLARS
to clarify expectations for student learning and to aid in evaluating
and revising the instruction
CULMINATING
Learning objectives for each lesson and mini-unit
should be derived directly from the language of the
Standards addressed by that activity.
MATHteaching in DYS schools
By the end of this mini-unit, students should:
KNOW (factual information, basic skills)
UNDERSTAND (big ideas, concepts)
And therefore be ABLE TO DO (final assessment, performance,
measurement of objectives)
58
REFLECTION
59
Stimulate student interest and motivation to participate
Students make meaning of content information and begin to
demonstrate, through ongoing assessment, what they know
and understand
Usually a final assessment in which students demonstrate their
level of achievement with regard to the learning objectives.
After the mini-unit is completed, make note of adjustments
you would make when using this unit again.
Massachusetts Math Curriculum Framework
STRANDS & STANDARDS
WHAT CAN YOU DO IN A DAY?
One of the basic conditions of teaching in DYS is that many of our students are with us for only a short period of time.Many
DYS youth are in detention sites, where they may spend only one or two days. While some teachers might want to tear
out their hair over this situation, DYS teachers can make a value of this necessity—not by repetitive skill drills or ditto
sheets, but by focusing on real math content through a
.
USING THE EMPHASIZED LEARNING STANDARDS
The Problem of the Day …
ensures that even in just
one day, you are helping
students learn something
that they can really use.
Select a lesson, activity or project you do that you think has strong mathematics
components.
Review the math standards above.
The Problem of the Day approach can be
differentiated to respond to students’ diverse
backgrounds, cognitive readiness and learning
styles, and is particularly useful in detention
settings, where student mobility may be extremely high. Teachers in assessment, short-term, long-term and communitybased diversion programs can also use a Problems of the Day to:
ENLIVEN a longer unit of study,
REPEAT what students have already studied, so they can practice and connect their new knowledge with what
Find 3 that are (or could be) supported by your lesson or activity
SIGNAL A CHANGE from one unit to another, or
Look over the math standards. Pick 1 standard that is particularly challenging for
your students.
they already know and can do.
Look through this manual; choose or brainstorm lessons, activities or projects that
you could do in your classroom to teach, reinforce, or extend this standard.
Problems of the Day are aligned with
of the curriculum framework, so
teachers in all settings can feel confident that they are using instructional time in ways that make a difference for their
students. Examples of interesting and engaging Problem of the Day techniques are provided in the next three sections of
this guide, which outline a wide array of curriculum resources for each strand, including:
MATHteaching in DYS schools
Pick 2 activities you could do in your programs to support that standard.
How will you assess whether or not students master the standard?
Emphasized Standards
Guidelines for Sequencing Instruction
Background Resources
Curriculum Resources
Problems of the Day
Integration of Number Sense and Operations
Sample Mini-unit
Connecting Math to Our Students’ Lives
Pulling it All Together
60
61
Massachusetts Math Curriculum Framework
STRANDS & STANDARDS
SEPTEMBER THROUGH NOVEMBER
Emphasized
Standards
65
Sequencing
Instruction
68
Background
Resources
for Teachers
72
Curriculum
Resources
for Teaching
76
Problems
of the Day
Curriculum
Resources
80
Integrating
Number Sense
and Operations
82
Sample
Mini-Unit
85
Connecting
Math to our
Students’ Lives
92
Pulling It All
Together
94
DATA ANALYSIS, STATISTICS & PROBABILITY
DATA, ANALYSIS, STATISTICS & PROBABILITY
WHAT DOES THIS STRAND COVER?
Formulate questions that can be addressed with data; collect, organize, and display relevant data to answer them
Select and use appropriate statistical methods to analyze data
WHAT ARE THE ESSENTIAL QUESTIONS?
In what ways can data analysis, statistics, and probability help clarify or answer questions?
In what ways can data analysis and probabilityhelp us predict and make decisions?
How do different categories or graphs influence the story the data tell?
What makes some ways of displaying data more useful or appropriate than others?
How do different statistical strategies tell stories in different ways?
What is “typical”?
What do we mean by “beating the odds”?
How do you know that a game is fair?
How do you know that something happens by chance?
Develop and evaluate inferences and predictions that are based on data
Understand and apply basic concepts of probability
WHAT STANDARDS SHOULD BE EMPHASIZED?
8.D.2
8.D.3
8.D.4
10.D.1
10.D.2
10.D.3
8.N.1
8.N.3
8.N.10
8.N.12
65
Select, create, interpret, and utilize various tabular and graphical representations of
data, e.g., circle graphs, Venn diagrams, scatterplots, stem-and-leaf plots, box-andwhisker plots, histograms, tables, and charts. Differentiate between continuous and
discrete data and ways to represent them
Find, describe, and interpret appropriate measures of central tendency (mean,
median, and mode) and spread (range) that represent a set of data. Use these notions
to compare different sets of data
Use tree diagrams, tables, organized lists, basic combinatorics (“fundamental
counting principle”), and area models to compute probabilities for simple
compound events, e.g., multiple coin tosses or rolls of dice
Select, create, and interpret an appropriate graphical representation
(e.g., scatterplot, table, stem-and-leaf plots, box-and-whisker plots, circle graph,
line graph, and line plot) for a set of data and use appropriate statistics
(e.g., mean, median, range, and mode) to communicate information about
the data. Use these notions to compare different sets of data
Approximate a line of best fit (trend line) given a set of data (e.g., scatterplot).
Use technology when appropriate
Describe and explain how the relative sizes of a sample and the population
affect the validity of predictions from a set of data.
Compare, order, estimate, and translate among integers, fractions and
mixed numbers (i.e., rational numbers), decimals, and percents
Use ratios and proportions in the solution of problems, in particular, problems
involving unit rates, scale factors, and rate of change
Estimate and compute with fractions (including simplification of fractions),
integers, decimals, and percents (including those greater than 100 and less than 1)
Select and use appropriate operations—addition, subtraction, multiplication,
division, and positive integer exponents—to solve problems with rational
numbers (including negatives)
DATA ANALYSIS, STATISTICS & PROBABILITY
Data is a powerful tool for survival and advocacy Data is a powerful tool for survival and advocacy Data is a powerful
tool for survival and advocacy
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helps support your point of view Data helps support your point of view Data Using data helps other people see things
Using data helps other people see things the way you do Using data helps other people see things the way you do Using
data helps other people see things the way you do Using Data always tells a story. Tell the story your way! Data always
Data always tells a story. Tell the story your way! Data always tells a story. Tell the story your way! Data always tells a
story. Tell the story your way! story. Tell the story your way! story. Tell the story your way! story. Tell the story your way!
Everyone should be able to create many kinds of graphs Everyone should be able to create many kinds of graphs Everyone
should be able to create many kinds of graphs should be able to create many kinds of graphs should be able to create
Everyone should be able to interpret many kinds of graphs Everyone should be able to interpret many kinds of graphs
Everyone should should be able to interpret many kinds of graphs Everyone should be able to interpret many kinds of
Statistics are useful tools for making decisions Statistics are useful tools for making decisions Statistics are useful tools
for making decisions Statistics are useful tools for making decisions Statistics are useful tools for making decisions
Statistics are useful tools for predicting outcomes Statistics are useful tools for predicting outcomes Statistics are useful
tools for predicting outcomes Statistics are useful tools for predicting outcomes Statistics are useful tools for predicting
Data is a powerful tool for survival and advocacy Data is a powerful tool for survival and advocacy Data is a powerful
tool for survival and advocacy
Data helps support your point of view Data helps support your point of view Data helps support your point of view Data
helps support your point of view Data helps support your point of view Data helps support your point of view
Using data helps other people see things the way you do Using data helps other people see things the way you do Using
data helps other people see things the way you do Using data helps other people see things the way you do
Data always tells a story. Tell the story your way! Data always tells a story. Tell the story your way! Data always tells a
story. Tell the story your way! Data always tells a story. Tell the story your way! Data always tells a story.
Everyone should be able to create many kinds of graphs Everyone should be able to create many kinds of graphs Everyone
should be able to create many kinds of graphs
Everyone should be able to interpret many kinds of graphs Everyone should be able to interpret many kinds of graphs
Everyone should be able to interpret many kinds of graphs Everyone should be able to interpret many kinds of graphs
Statistics are useful tools for making decisions Statistics are useful tools for making decisions Statistics are useful tools
for making decisions Statistics are useful tools for making decisions Statistics are useful tools Statistics are useful
Statistics are useful tools for predicting outcomes Statistics are useful tools for predicting outcomes Statistics are useful
tools for predicting outcomes Statistics are useful tools for predicting outcomes Statistics are useful tools for predicting
Data is a powerful tool for survival and advocacy Data is a powerful tool for survival and advocacy Data is a powerful
tool for survival and advocacy Data is a powerful tool for survival and advocacy Data is a powerful tool for survival
MATHteaching in DYS schools
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Data is a powerful tool for survival and advocacy Data is a powerful tool
Data is a powerful tool for survival and advocacy Data is a powerful
Everyone should be able to create many kinds of graphs
DATA ANALYSIS, STATISTICS & PROBABILITY
A.
SEQUENCING INSTRUCTION IN DYS
Each strand of the Math Curriculum Framework
is taught during the same three-month period
of time each year.
B.
Each strand of the Math Curriculum Framework
is broken down into discrete topics that address
all of the Emphasized Standards for that strand.
.
COLLECTING AND
DISPLAYING DATA
C.
To teach each strand properly, teachers must
focus on every topic in the strand. The type of
setting and the stability or mobility of students
determines how teachers proceed through all
topics (depicted visually on the following pages).
MATHteaching in DYS schools
September through November
December through March
April through June
Data Analysis, Statistics and Probability
Patterns, Relations, and Algebra
Geometry and Measurement
Following the calendar ensures that (1) all crucial math information will be addressed, and (2) students
transitioning between DYS settings will be exposed to all elements of a unified math curriculum
TRENDS AND
CHANGE OVER TIME
MEASURES OF
CENTRAL TENDENCY
PROBABILITY
PULLING IT ALL
TOGETHER
In treatment settings, teachers should plan to spend between one and two weeks teaching a MiniUnit on each topic. After proceeding through all topics in the strand, they should then cycle through
the topics again, using different materials, lessons, and examples. By way of contrast, teachers in
very short-term settings should proceed through all strands by spending just one day addressing
each topic, and then cycling through the topics again and again from September through November.
Teachers in all settings must plan carefully to integrate number sense topics and all other topics in
the Data Analysis, Statistics and Probability strand.
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DATA ANALYSIS, STATISTICS & PROBABILITY
SEQUENCING INSTRUCTION IN DYS
ADDRESSING ALL TOPICS IN
DIFFERENT DYS SETTINGS
In classes with very high mobility,
address each topic through Problems
of the Day. Examples are provided
later in this section.
m
r
e
t
t
r
o
Sh
l e ms o
ASSESSMENT SETTINGS
In most settings and classrooms,
instruction should proceed straight
through all topics in the strand.
TREATMENT or OTHER
LONGER-TERM SETTINGS
In treatment settings, instruction
should cycle through all topics, then
repeat in order, incorporating different lessons and examples each time.
ALL DYS SETTINGS
In all settings, Problems of the Day
can be used to enliven a longer unit
of study, signal a change from one
unit to another, or enable students to
apply, deepen, or connect their new
knowledge to what they already
know, understand, and can do.
MATHteaching in DYS schools
ng,
COLLECTING AND
DISPLAYING DATA
In
TRENDS AND
CHANGE OVER TIME
tre
atm
ent
se
may be appropriate in setting
”Problems of the D
ay”
MEASURES OF
CENTRAL TENDENCY
s with ve
ry high
mobil
i ty
PULLING IT ALL
TOGETHER
PROBABILITY
m
xa
e
d
s an
n
o
less
new
h
t
i
ew
he cycl
t
t
a
e
p
e
r
n
each topic, the
pl
es
DETENTION SETTINGS
“ Prob
l as s l o
ust one c
j
”
,
y
a
f th e D
INTEGRATE
NUMBER SENSE AND
OPERATIONS
INTO ALL
MATHEMATICS
TOPICS, UNITS
& STRANDS
ttin
gs, p
roceed
with
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DATA ANALYSIS, STATISTICS & PROBABILITY
BACKGROUND/RESOURCES FOR TEACHERS
These resources provide background knowledge for teachers. They are not student materials, but may help DYS teachers
PRIMARY RESOURCES
GETTING READY TO TEACH THIS STRAND
prepare to teach a unit on Data Analysis, Statistics, and Probability more effectively and comfortably
WHAT THE RESOURCE OFFERS TEACHERS PREPARING TO TEACH THIS STRAND
CHAPTERS AND/OR PAGES
Standards for School Mathematics, chapter 3, pages
NCTM (National Council of Teachers of Mathematics) 48-51: Data Analysis and Probability Standards
Principles and Standards for School Mathematics
Grades 6-8 in Chapter 6 on pages 248-255
Grades 9-12 in Chapter 7 on pages 324-333
Overview of the main concepts and skills NCTM suggests be taught and learned in Statistics and
Probability Examples of what these ideas look like in the classroom
Designing and Implementing Mathematics Instruction
Chapter 15, pages 536-586
for Students with Diverse Learning Needs
Overview of how the ideas of data, analysis, statistics, and probability develop in Grades 2-8. Many DYS students
have missed opportunities for basic concepts to solidify, and may therefore benefit from returning to
earlier stages of concept development
Rethinking Mathematics:
Teaching Social Justice by the Numbers
Also see the Rethinking Mathematics website at
http://www.rethinkingschools.org/publication/math/
RM_math.shtml
Collection of 30 articles and examples about weaving social justice issues throughout the math curriculum;
additional source and supplemental materials for lessons on the associated website
Many Points Make a Point: Data and Graphs
(Teacher Book)
Unit Introduction, pages xxi-xxv
Describes the approach to developing important ideas when working with data and graphs
Authentic Learning Activities in
Middle School Mathematics:
Data Analysis, Statistics, and Probability
Pages 6-18
E RESOURCE
Philosophy and rationale Discussion of what’s new in teaching Data Analysis, Statistics, and Probability
ONLINE RESOURCES
URL
WHAT THE WEBSITE OFFERS TEACHERS PREPARING TO TEACH THIS STRAND
“Illuminations” on the NCTM website
http://illuminations.nctm.org/WebResourceList.aspx?
Ref=2&Std=4&Grd=0
Provides many online activities for students, for example, Circle Grapher
“Illuminations” on the NCTM website
http://illuminations.nctm.org/ActivityDetail.aspx?
ID=146
Provides many online activities for students, for example, an Applet for plotting points and finding a line of best fit
LINCS Science and Numeracy Collection
http://literacynet.org/sciencelincs/slnum-data.html
Web-based activities for learners to explore statistics and probability
MATHteaching in DYS schools
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DATA ANALYSIS, STATISTICS & PROBABILITY
TEACHERS ENTER OTHER RESOURCES HERE
PRIMARY RESOURCES
NOTES ON THIS RESOURCE
CHAPTERS AND/OR PAGES
Facing the Odds: The Mathematics of Gambling and
Other Risks by Harvard Medical School's Division on
Addictions, and the Massachusetts Council on
Compulsive Gambling
ONLINE RESOURCES
MATHteaching in DYS schools
ADDITIONAL RESOURCES
Helpful explanations and case studies on probability, statistics, and the lottery Curriculum aims to make
mathematics more meaningful by introducing concepts of probability and statistics through the use of gambling and
media-related topics Helps students develop critical thinking and number sense skills, and apply these skills to
media, advertising, and gambling issues
URL
NOTES ON THIS RESOURCE
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DATA ANALYSIS, STATISTICS & PROBABILITY
CURRICULUM RESOURCES FOR TEACHING
PRIMARY
CURRICULUM
RESOURCES
EMPower:
Many Points Make a Point
Bridges to Algebra and
Geometry: Mathematics in
Context (2004, 2nd edition)
Number Power:
Analyzing Data
LESSONS AND TOPICS IN THE DATA ANALYSIS, PROBABILITY &
THE TOPICS IN THIS STRAND
STATISTICS STRAND PROCEED DEVELOPMENTALLY SEPTEMBER–NOVEMBER
COLLECTING AND
DISPLAYING DATA
TRENDS AND
CHANGE OVER TIME
MEASURES OF
CENTRAL
TENDENCY
Opening through
Lesson 4
Lessons 5 though 9
Lessons 5 though 9
Chapter 2: Teacher Resource Book,
pages 57– 106
Chapter 2: Teacher Resource Book, pages 57–106
Student Book, pages 68-125
PROBABILITY
Lesson 10 and Closing
Ratio and Proportion:
Chapters 6.1-6.3
Student Book, pages 68-125
Probability:
Chapters 6.4-6.8
Pages 8-91
Pages 92-103
Pages 8-91
EMPower
Number Power 2
MATHteaching in DYS schools
Pages 105-165
Data Analysis & Statistics: Unit 1
“Will the African Elephant Become
Extinct in Your Lifetime?”
Authentic Learning
Activities in Middle School
Maths: Data Analysis,
Statistics, & Probability
(2002 edition)
Discovering Algebra: An
Investigative Approach
PULLING IT ALL
TOGETHER
Probability: Unit 2
“Is the World Series Rigged?”
Lessons 1.1-1.7
Lessons 1.1-1.7 (continues to following topic)
See page 83 for curriculum resources to integrate
76
Proportional Reasoning:
Lessons 2.1-2.2
Probability:
Lessons 2.6 and 2.7
Number Sense & Operations into instruction in the Data Strand
77
DATA ANALYSIS, STATISTICS & PROBABILITY
TEACHERS ENTER OTHER RESOURCES HERE
TEACHERS-SELECTED
RESOURCES
LESSONS AND TOPICS IN THE DATA ANALYSIS, PROBABILITY &
COLLECTING AND
DISPLAYING DATA
ADDITIONAL RESOURCES
STATISTICS STRAND PROCEED DEVELOPMENTALLY SEPTEMBER–NOVEMBER
MEASURES OF
CENTRAL
TENDENCY
TRENDS AND
CHANGE OVER TIME
PROBABILITY
PULLING IT ALL
TOGETHER
Resources for integrating
Number Sense & Operations
into this strand
MATHteaching in DYS schools
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DATA ANALYSIS, STATISTICS & PROBABILITY
SHORT-TERM PROBLEMS OF THE DAY
CURRICULUM RESOURCES
THESE SHORT 1-DAY MODULES CAN BE DIFFERENTIATED FOR STUDENTS’ VARIED STRENGTHS
COLLECTING, ANALYZING AND
DISPLAYING DATA
“Poverty and World Wealth”
helps students use estimation and proportional
reasoning to think about where the world’s
population and wealth are located; see Rethinking
Mathematics, page 64 -67, and go to the website
(http://www.rethinkingschools.org/publication/math/
RM_math.shtml) for handouts and worksheets for
students to record their estimates of world
population and wealth by continent
“The Challenge”
uses football team statistics to help students
determine averages; see Math Stories for
Problem-Solving Success, pages 161-164
SHORT MODULES MAY BE PARTICULARLY USEFUL IN DETENTION PROGRAMS
MEASURES OF
CENTRAL TENDENCY
TRENDS AND CHANGE OVER TIME
“Sketch This”
helps students look holistically at trend lines and translate
interesting stories into line graphs; see
EMPower: Data and Graphs, Lesson 5
“Roller Coaster Rides”
engages students in making statements about graphs, and then
judging the veracity of other students’ statements; see
EMPower: Data and Graphs, Lesson 6
“A Mean Idea”
asks students to use a variety of strategies to think
about the mean as a measure of the center of the data;
see EMPower, Data and Graphs, Lesson 7
"Median"
asks students to “act out” the median;
see EMPower, Data and Graphs, Lesson 9
“Survey”
“Mystery Cities”
can be modified with questions that are appropriate
is also a good US geography lesson, as students try to match
to the interests of students in your classroom
descriptions of cities with graphs; see
(e.g., “How many music CDs did you buy this year?”
EMPower:
Data and Graphs, Lesson 8
1, 2, 3, 4, 5, or “other”); see Math Stories for
Problem-Solving Success, pages 188-191
“Around the World”
facilitates cooperative problem solving with
descriptive statistics; see
Get It Together, pages 152-157
“Group Project”
is an exercise in collecting, displaying, and
analyzing data. Methods for data collection can be
adapted in DYS facilities by asking between 15-20
students in other classes, as well as the students in
your class, to complete the surveys on page 26, and
giving each pair one set of surveys; see Number
Power: Analyzing Data, pages 26, 60, and 90
MATHteaching in DYS schools
Interpreting Line Graphs
Number Power: Analyzing Data
pages 38-43
PROBABILITY
“What Is the Average?”
explores mean, median, and mode; see
Math Stories for Problem-Solving Success,
pages 157-160
Mean, Median, Mode
reviews the three measures of central tendency
and asks students to think critically about them;
see Number Power: Analyzing Data
pages 54-59
Analyzing Trends
Number Power: Analyzing Data
pages 86-89
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81
“Which Spinner Is It?”
and
“Draw the Spinner”
present cooperative problem-solving using the
properties of spinners; see Get It Together
pages 100-105 and 106-111
“Take a Chance”
enables students to determine the probability
of winning by demonstrating the properties
of a spinner wheel; see Math Stories for Problem
Solving Success, pages 98-101
“Tickets”
helps students determine the probability of winning
when buying raffle tickets; see Math Stories for
Problem Solving Success, pages 75-78
“Lucky Seven”
guides students to determine the probability of
rolling a seven (and other outcomes)
with two dice, and explores experimental probability
vs. theoretical probability; see Math Stories for
Problem Solving Success, pages 114-117
DATA ANALYSIS, STATISTICS & PROBABILITY
INTEGRATING
INTEGRATING
NUMBER SENSE
and
OPERATIONS
into the
DATA ANALYSIS, STATISTICS
AND PROBABILITY
STRAND
EMPHASIZED STANDARDS
8.N.1
Compare, order, estimate, and translate among integers, fractions and
mixed numbers (i.e., rational numbers), decimals, and percents
to integrate into this strand
Connections between fractions,
decimals, and percents and
their comparative values
Ratio and proportion
applications
How and when
to add, subtract, multiply,
and divide with fractions,
decimals, and percents
STANDARD 8.N.1
STANDARD 8.N.3
STANDARDS
8.N.10 and 8.N.12
Basic level
Understanding the “benchmark”
fractions, decimals, and percents
(halves, quarters, tenths,
hundredths and their decimal and
percent equivalents)
Basic level
Understanding how to reason
about ratios and whether or not
things are proportional
Resources
EMPower: Using Benchmarks:
Fractions, Decimals and Percents
(Lessons 1- 6)
8.N.3
Use ratios and proportions in the solution of problems, in particular,
problems involving unit rates, scale factors, and rate of change
EMPower: Split It Up: More
Fractions, Decimals, and Percents
(Lessons 1-8)
8.N.10
Resources
EMPower: Reasoning With Ratios:
Keeping Things in Proportion
(Lessons 1-4)
EMPower: Operation Sense: Even
More Fractions, Decimals,
and Percents
(Lesson 1)
Estimate and compute with fractions (including simplification of fractions), integers,
decimals, and percents (including those greater than 100 and less than 1)
8.N.12
Select and use appropriate operations—addition, subtraction, multiplication, division, and positive integer
exponents—to solve problems with rational numbers (including negatives).
Integrating these NUMBER SENSE AND OPERATIONS standards into the DATA ANALYSIS, STATISTICS AND
PROBABILITY STRAND helps students develop a sense of “benchmarks” for good estimates about the relative size of
data categories. (Students can use calculators to get more precise.) Proportional reasoning undergirds students’ abilities
to work with statistics and probability., and will help students become more successful in these areas. Finally (and most
importantly), when taught alone, Number Sense and Operations can become unnecessarily dry and repetitive. Incorporating
work on Number Sense and Operations into other areas of mathematics can make these fundamental skills come alive!
MATHteaching in DYS schools
NUMBER SENSE and OPERATIONS TOPICS
NUMBER SENSE
82
Advanced
Understanding how to work with
the “messier numbers,” being able
to compare and order numbers
such as 1⅞, 0.463, and 1.5%
Resources
Number Power 2, pages 11-21,
62-71, and 98-103
Number Power 2
Pre-algebra, pages 10-15
83
Advanced
Understanding how to work
with the cross-product rule, and
why the rule works
Resources
Number Power 2
Pre-algebra, pages 43-53
Discovering Algebra: An
Investigative Approach,
chapters 2.1 and 2.2
Basic level
Understanding how to add,
subtract, multiply, and divide with
the “benchmark” fractions,
decimals, and percents; for
example, adding 4 ½ + 7 ¾,
multiplying 1.5 by 10,000, or
determining 40% of 60. Emphasis
on the meaning of the operations
Resources
EMPower: Operation Sense:
Even More Fractions, Decimals
and Percents
(Lessons 1- 6)
Advanced
Understanding how to estimate
and compute with the “messier
numbers”; for example, adding
3 ⅞ + 4.56, or determining
5.6% of 64,300.
Resources
Number Power 2
Selected pages from:
Fractions, pages 8-58
Decimals, pages 60-94
Percents, pages 96-121
DATA ANALYSIS, STATISTICS & PROBABILITY
Look over the standards and suggested topics for Integrating Number Sense and Operations. Choose one standard
that you are ready to teach in your classroom.
What resources would you use to teach this standard through stand-alone lessons?
What kinds of preparation would it require? What kinds of results would you expect?
SA
MP
L
EM
IN
U
I-
THIS STRAN
R
O
F
D
T
I
N
What resources would you use to teach this standard integrated into Data Analysis, Statistics and Probability?
What kinds of preparation would it require? What kinds of results would you expect?
Keep notes about how you teach Number Sense and Operations. What works well, and what doesn’t work so well?
Reflect on how your approach affected students’ skill acquisition, their engagement in the classroom, and your
ability to differentiate instruction.
PRE-ASSESSMENT
How will you determine students’ readiness for this unit? What data will you collect?
What survey of prerequisite learning (i.e., KWL charts, journal prompts, oral surveys)
will you use)?
RESOURCES
Be sure to consider differences in students’ reading levels, interests, readiness to learn
(prerequisite learning), learning styles (multiple intelligences: audio, visual, kinesthetic, etc.)
and backgrounds when selecting culturally responsive resources for the mini-unit.
STAGES OF LESSONS WITHIN THE MINI-UNIT
Introductory Stimulate student interest in the topic, motivate
students to participate in the project
Instructional Students make meaning of content information
and begin to demonstrate, through ongoing
assessment, what they know and understand
Culminating Usually a final assessment, in which students
demonstrate their level of achievement with
regard to the learning objectives
MATHteaching in DYS schools
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DATA ANALYSIS, STATISTICS & PROBABILITY
DATA ANALYSIS, STATISTICS & PROBABILITY
Learning objectives in this miniunit are tied to the following:
SAMPLE MINI-UNIT
5-8 days
DESIGNER’S NAME
DESIGNER’S EMAIL
Mary Jane Schmitt
Data Analysis, Statistics, and Probability
Collecting, Displaying, and Analyzing Data
LEARNING OBJECTIVES
By the end of this mini-unit, students should:
KNOW…
8.D.2
mary_jane_ [email protected]
STRAND
MINI-UNIT TOPIC
Select, create, interpret, and utilize various tabular and
graphical representations of data, e.g., circle graphs, Venn
diagrams, scatterplots, stem-and-leaf plots, box-and-whisker
plots, histograms, tables, and charts. Differentiate between
continuous and discrete data and ways to represent them.
10.D.1
How to construct frequency graphs, bar graphs, and
circle graphs from raw data
Select, create, and interpret an appropriate graphical
representation (e.g., scatterplot, table, stem-and-leaf plots,
box-and-whisker plots, circle graph, line graph, and line plot)
for a set of data, and use appropriate statistics (e.g., mean,
median, range, and mode) to communicate information about
the data. Use these notions to compare different sets of data.
How to interpret the story the graphs tell and make
numerical statements about the data,using benchmark
fractions, decimals, and percents
How to compare data from various samples
UNDERSTAND…
8.N.10
How categories affect the stories that data tell
(from Number Sense and Operations strand)
Compare, contrast, and connect graphic
representations of data
Estimate and compute with fractions (including
simplification of fractions), integers, decimals, and percents
(including those greater than 100 and less than 1).
How graphs convey information, help make
decisions and predictions, and pose questions
…and therefore be able to
DO
MATHteaching in DYS schools
EMPHASIZED
STANDARDS
Given a set of data, students will organize (categorize),
display graphically, and make conclusions and
decisions about the data
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DATA ANALYSIS, STATISTICS & PROBABILITY
MINI-UNIT: COLLECTING, DISPLAYING, AND ANALYZING DATA (continued)
OUTLINE OF LESSONS
PRE-ASSESSMENT
“Countries in Our Closets”
RESOURCES
Lesson tasks and activities to support students’ achievement of learning objectives
Students will engage in an activity in EMPower: Many
Points Make a Point: Data and Graphs. “Opening the
Unit” provides them with the opportunity to examine and
categorize a set of 12 graphs, and select one graph from
which to tell a story. They will also show how facile they
are with common “benchmark” fractions, decimals, and
percents by taking the Initial Assessment in EMPower
Split It Up: More Fractions, Decimals, and Percents. The
purpose of the assessment is to welcome students to the
study of data.
EMPower Many Points Make a Point: Lesson 1
Countries in Our Closets. Teacher Book , pages 15-26;
Student Book, pages 7-18
Introductory
In this lesson, student pairs have a set of data (30 pieces
of clothing with the labels showing where the clothing is
manufactured). First, they create a frequency graph (by
country), and then reorganize the data by continent. They
make numerical statements and compare their data with
another set of data.
For assessments, please refer to Many Points Make a
Point, Teacher Book, pages 1-14; Student Book, pages
1-4, Many Points Make a Point Teacher Book, pages
141-144, and Split It Up Teacher Book, pages 127-131
NOTE: This lesson needs a slight adaptation to work well
in a DYS setting. The original lesson asks students to go
home, look in their closets, select about 8 pieces of
clothing, and bring that data into class. Since this is not
possible, we suggest you give each pair of students an
envelope with pictures of 30 pieces of clothing from the
GAP website, marked with the country that is on the
label. This information is easily available at the
company’s website; go to:
EMPower: Many Points Make a Point: Data and Graphs
EMPower: Split It Up: More Fractions, Decimals, and
Percents
MATERIALS
LESSON ONE
“
http://www.gapinc.com/public/SocialResponsibility/sr_ethic_where.shtml
Newsprint (one sheet per pair of students)
Copies of graphs (see pre-assessment)
Index cards, Calculators
Excel software (if available)
One-centimeter graph paper
Markers, Tape, Rulers
For Lesson 1adaptation : Envelopes with pictures of 30
pieces of clothing (from the GAP) indicating country
of manufacture
LESSON TWO
“Most of Us Eat…”
EMPower Many Points Make a Point: Lesson 2:
Teacher Book , pages 27-38; Student Book, pages 19-35.
Instructional (optional)
This lesson gives students more experience generating
data, choosing their own categories, and using fractions
and percents to describe the data. They start out collecting
data on frequently eaten foods, pay attention to the need
for consistent categories, and practice describing the data.
NOTE: You might want to adapt the lesson by choosing a
topic of particular interest to your students, e.g., favorite
cars, favorite music artists, etc.
MATHteaching in DYS schools
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DATA ANALYSIS, STATISTICS & PROBABILITY
MINI-UNIT: COLLECTING, DISPLAYING, AND ANALYZING DATA (continued)
LESSON THREE
Instructional
This lesson uses real MCAS problems that relate to
categorical data and graphs that students have been
investigating throughout the lessons in this mini-unit.
“Displaying Data In New Ways”
Instructional
EMPower Many Points Make a Point: Lesson 3
Teacher Book pages 39-47; Student Book pages 35-46
2005 Mathematics Grade 10
Question 18: Short Answer
2005 Mathematics Grade 10
Question 23: Multiple Choice
2005 Mathematics Grade 10
Question 37: Multiple Choice
http://www.doe.mass.edu/mcas/search/
“A Closer Look At Circle Graphs”
2005 Mathematics Grade 10 Question 42:
Open Response
EMPower Many Points Make a Point: Lesson 4
Teacher Book pages 49-59; Student Book pages 47-62
2004 Mathematics Grade 10 Question 13:
Multiple Choice
Students construct circle graphs to show how the parts
relate to the whole. They apply knowledge of
“benchmark” percents to estimate the size of the slices of
the pie, and interpret several circle graphs.
LESSON FIVE
Culminating
Culminating
“Suggested Choices from real MCAS Questions”
This lesson uses the frequency graphs and data from
Lesson 1 or 2. Students do a hands-on activity in which
they transform frequency graphs into bar graphs and then
into circle graphs. They contrast the two formats,
considering how the graphs are alike and different.
LESSON FOUR
LESSON SIX
2004 Mathematics Grade 10 Question 29: Multiple Choice
Have students read the information provided in each MCAS
problem and discuss how these problems are similar to the
work in their previous lessons.
“Midpoint Assessment: The Data Say”
EMPower Many Points Make a Point: Teacher Book
pages 61-70; for students, photocopy pages 147-153.
Students apply what they have learned in Lessons 1-4.
Have students choose one of the problems for the class to
work on together at the board. Be sure to demonstrate how
they should clearly write their solutions and their work.
…Model
Using an overhead if possible, show the exemplars that can
be found online, at least for 1-point, 2-point, and 4-point
questions.
…Wrap-Up
Give students time to work on the rest of the problems,
either independently or in a small group. For the 2005
Mathematics Grade 10 Question 42: Open Response
question, ask them to write their solutions so someone else
would be able to understand their thinking.
NOTE: You might want to adapt the lesson by choosing a
topic of particular interest to your students, for example,
favorite cars, music artists, or other performers.
Another source of data that may be responsive to
students’ interests is the website below, which lists the
many variations that consumers can choose when they
design custom Timberland boots:
…Launch
…Practice
http://www.timberland.com/customboots/index.jsp?clickid=topnav_boots_img
MATHteaching in DYS schools
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DATA ANALYSIS, STATISTICS & PROBABILITY
CONNECTING MATH TO OUR STUDENTS’ LIVES Sample Lessons and Resources
One of the most fundamental ways to connect with others is through our names. How are they similar? How are they
different? What personal, family, and community stories do they convey? Using students’ own names as a source of
data offers an accessible and engaging means to explore various ways to that data can be presented and analyzed.
Invite students to write down the name of every person in the room, including teachers, students, and line staff. Next
to each person’s name, students should indicate how many letters are in the name. Use this information as data to
construct a
that students can fill in to measure how often each possible answer occurs.
Following is an example of a simple Frequency Table:
Number of letters
in each person’s
first name
3
4
5
6
7
7
9
10
Next, students can construct a
from the information in the Frequency Table, and describe the
distribution of the dots (i.e., What shape do they make? How are they spread out? Are they symmetrical or
asymmetrical?)
After creating a
Dot-Plot, students
can then proceed to
construct a
11
10
Total Number of People
9
8
7
6
5
4
3
2
1
0
2
3
4
MATHteaching in DYS schools
5
6
7
8
9
Number of Letters in First Name
10
11
12
In a further lesson, students can use the data from the Frequency Table to create a
by grouping names
into clusters of multiple lengths (note: in all cases, it is important to label the x and y axes of the chart clearly, indicating
the number of people and the number of letters in their names or the number of letters in the names that are grouped in
each category).The basic frequency data derived from students’ names can be used to create a wide range of tabular and
graphical representations of statistical information, including dot-plots, bar charts, histograms, circle graphs, pie charts,
Venn diagrams, scatterplots, stem-and-leaf plots, box-and-whisker plots, tree diagrams, and more.
Number of letters in each name
3, 4, or 5
letters
6, 7, or 8
letters
9, 10, or more
letters
Number of people
Number of people
12
CONNECTING MATH TO OUR STUDENTS’ LIVES
in which, instead of
using dots, students
draw bars with
different heights to
represent the
frequency of each
response. Questions
about the stories the
bar chart tells—
How do the names
cluster? Is the
distribution of long
and short names
even?—can be very
engaging when the
data derive from
students’ own lives.
92
A great source for additional data sets and scenarios that respond to students’ lives can be found at www.radicalmath.org,
where lessons and source materials include military recruitment, racial profiling, playing the lottery, statistics on
smoking, labor, health, health insurance, and gentrification, and trends in poverty, immigration, college funding,
incarceration, and demographics. For example, the Radical Math site links to the following “Military Math” problem,
which engages students in the Essential Question:
MILITARY MATH
Month
July
August
September
October
November
December
January
February
March
April
May
June
Casualties
55
85
54
96
88
67
65
56
33
81
71
61
http://www.globalsecurity.org/military/ops/iraq_casualties.htm
93
The chart on the left lists the number of American soldiers
who died in Iraq each month between July of 2005 and
June of 2006.
Students should answer the following questions on the
basis of these casualty figures:
1. First, imagine you work for the army, and you need to
put out a press release that states the average number
of soldiers killed per month. What number would you
choose as your average? Explain how you got this
number, and why you chose this as your method.
2. Next, imagine you are working for an anti-war
organization, and you need to put out a press release
that states the average number of soldiers killed per
month. What number would you choose as your
average? Explain how you got this number, and
why you chose this as your method.
3. Were the two averages you chose the same, or different?
Explain why.
DATA ANALYSIS, STATISTICS & PROBABILITY
“PULLING IT ALL TOGETHER”
USING MCAS RESOURCES EFFECTIVELY
Whether DYS teachers are using integrated lessons, mini-units, or
Problems of the Day, you can find MCAS released questions
that help students review, practice, and apply their learning
in this strand.
LAUNCH
MODEL
The URL on the right will take you to the Massachusetts
Department of Education’s MCAS Question Search Tool.
PRACTICE
This tool allows users to search real questions that have been used
in previous years’ MCAS exams. Teachers do not have to
register to use the MCAS Question Search Tool.
WRAP-UP
http://www.doe.mass.edu/mcas/search/
Have students choose one of the problems for the class to work on together at the
board. Be sure to demonstrate how they should clearly write their solutions and
their work.
Give students time to work on the rest of the problems, either independently or
in small groups. For Open Response questions, ask students to write their
solutions so that another person would be able to understand their thinking.
Using an overhead if possible, show the exemplars that can be found online,
at least for 1-point, 2-point, and 4-point questions.
RECOMMENDED MCAS QUESTIONS BY TOPIC, YEAR, GRADE AND TYPE
Question 18
Question 23
Question 37
Question 42
Question 13
Question 29
Short Answer
Multiple Choice
Multiple Choice
Open Response
Multiple Choice
Multiple Choice
TRENDS AND
CHANGE OVER TIME
2005 Math Grade 10 Question 2
2005 Math Grade 10 Question 14
2003 Math Grade 8 Question 3
Multiple Choice
Multiple Choice
Multiple Choice
MEASURES OF
CENTRAL TENDENCY
2005 Math Grade 10
2005 Math Grade 10
2005 Math Grade 10
2004 Math Grade 10
Question 22
Question 28
Question 30
Question 31
Multiple Choice
Multiple Choice
Multiple Choice
Open Response
PROBABILITY
2004 Math Grade 10
2004 Math Grade 8
2003 Math Grade 10
2004 Math Grade 8
Question 12
Question 20
Question 10
Question 33
Multiple Choice
Short Answer
Multiple Choice
Multiple Choice
COLLECTING AND
DISPLAYING DATA
Teachers can search released MCAS questions according to
the following criteria, which are available through “pull-down menus” on this website:
YEAR
Choose the year in which test items were administered
GRADE
Choose the grade level associated with test items
SUBJECT AREA/QUESTION CATEGORY
Choose the subject area of the test in which items were administered (i.e., Mathematics)
and by the question category (i.e., the strand; in this case, Data Analysis, Statistics and Probability)
QUESTION TYPE
Choose multiple choice, open response, short answer, or writing prompt questions
Each of the MCAS math questions will also indicate clearly (using a graphic of a calculator, with or without a large red
X) whether students were allowed to use a calculator for this question when taking the
MCAS. Lessons that include MCAS released items should follow the same guidelines.
MATHteaching in DYS schools
Have students read the information provided in each MCAS problem and discuss
how these problems are similar to their work in previous lessons.
94
95
2005 Math Grade 10
2005 Math Grade 10
2005 Math Grade 10
2005 Math Grade 10
2004 Math Grade 10
2004 Math Grade 10
DATA ANALYSIS, STATISTICS & PROBABILITY
DECEMBER THROUGH MARCH
Emphasized
Standards
99
Sequencing
Instruction
102
Background
Resources
for Teachers
106
Curriculum
Resources
for Teaching
110
Problems
of the Day
Curriculum
Resources
116
Integrating
Number Sense
and Operations
118
Sample
Mini-Unit
121
Pulling It All
Together
129
Connecting
Math to Our
Students’ Lives
130
PATTERNS, RELATIONS & ALGEBRA
PATTERNS, RELATIONS & ALGEBRA
WHAT DOES THIS STRAND COVER?
Understanding patterns, relations, and functions
Representing and analyzing mathematical situations and structures using algebraic symbols
Using mathematical models to represent and understand quantitative relationships
Analyzing change in various contexts
WHAT ARE THE ESSENTIAL QUESTIONS?
How do you notice patterns?
How can you describe them?
In what ways do patterns predict what will happen in the future?
In what ways do patterns show what is changing in a situation?
In what ways do patterns show why something is changing in a situation?
How can I use patterns to show a relationship?
How can I find the values that make a relationship true?
How can I compare patterns or relationships to make the best choice?
What makes some ways of representing a relationship better or more appropriate than others?
WHAT STANDARDS SHOULD BE EMPHASIZED?
8.P.1
8.P.2
8.P.4
8.P.5
8.P.6
8.P.7
8.P.9
8.P.10
10.P.1
10.P.2
10.P.6
10.P.7
99
Extend, represent, analyze, and generalize a variety of patterns with tables, graphs, words,
and, when possible, symbolic expressions. Include arithmetic and geometric progressions,
e.g., compounding.
Evaluate simple algebraic expressions for given variable values, e.g., 3a2 – b for a = 3 and b = 7
Create and use symbolic expressions and relate them to verbal, tabular, and graphical
representations
Identify the slope of a line as a measure of its steepness and as a constant rate of change from
its table of values, equation, or graph. Apply the concept of slope to the solution of problems
Identify the roles of variables within an equation, e.g., y = mx + b, expressing y as a function
of x with parameters m and b
Set up and solve linear equations and inequalities with one or two variables, using
algebraic methods, models, and/or graphs
Use linear equations to model and analyze problems involving proportional relationships;
use technology as appropriate
Use tables and graphs to represent and compare linear growth patterns. In particular,
compare rates of change and x- and y-intercepts of different linear patterns.
Describe, complete, extend, analyze, generalize, and create a wide variety of patterns,
including iterative, recursive (e.g., Fibonacci Numbers), linear, quadratic, and
exponential functional relationships.
Demonstrate an understanding of the relationship between various representations of a line.
Determine a line’s slope and x- and y-intercepts from its graph or from a linear equation that
represents the line. Find a linear equation describing a line from a graph or a geometric
description of the line, e.g., by using the “point-slope” or “slope y-intercept” formulas.
Explain the significance of a positive, negative, zero, or undefined slope
Solve equations and inequalities including those involving absolute value of linear
expressions (e.g., |x - 2| > 5) and apply to the solution of problems
Solve everyday problems that can be modeled using linear, reciprocal, quadratic, or exponential
functions. Apply appropriate tabular, graphical, or symbolic methods to the solution. Include
compound interest, and direct and inverse variation problems. Use technology when appropriate
PATTERNS, RELATIONS, & ALGEBRA
Patterns can be used to make predictions and arguments. Patterns can be used to make predictions and arguments.
Patterns can be used to make predictions and arguments. Patterns can be used to make predictions and arguments
There are many ways to represent mathematical situations. Which ways work best to tell your story? There are many
ways to represent mathematical situations. Which ways work best to tell your story? There are many ways to represent
By generalizing relationships you can understand them more deeply. By generalizing relationships you can understand
Using and analyzing mathematical models helps you solve complicated problems. Using and analyzing mathematical
Math is a powerful tool to understand and describe how things change. Math is a powerful tool to understand and describe
them more deeply. By generalizing relationships you can understand them more deeply. By generalizing relationships
models helps you solve complicated problems. Using and analyzing mathematical models helps you solve complicated
how things change. Math is a powerful tool to understand and describe how things change. Math is a powerful tool to
Everyone can learn the language of algebra. Everyone can learn the language of algebra. Everyone can learn the language
of algebra. Everyone can learn the language of algebra. Everyone can learn the language of algebra. Everyone can learn
You can use patterns to compare relationships and help make the best choice. You can use patterns to compare relation-
ships and help make the best choice. You can use patterns to compare relationships and help make the best choice. You
Algebra helps you solve real-life problems quickly. Algebra helps you solve real-life problems quickly. Algebra helps
Patterns can be used to make predictions and arguments. Patterns can be used to make predictions and arguments.
By generalizing relationships you can understand them more deeply. By generalizing relationships you can understand
There are many ways to represent mathematical situations. Which ways work best to tell your story? There are many
Using and analyzing mathematical models helps you solve complicated problems. Using and analyzing mathematical
Math is a powerful tool to understand and describe how things change. Math is a powerful tool to understand and describe
you solve real-life problems quickly. Algebra helps you solve real-life problems quickly. Algebra helps you solve real-
Patterns can be used to make predictions and arguments. Patterns can be used to make predictions and arguments
them more deeply. By generalizing relationships you can understand them more deeply. By generalizing relationships
ways to represent mathematical situations. Which ways work best to tell your story? There are many ways to represent
models helps you solve complicated problems. Using and analyzing mathematical models helps you solve complicated
how things change. Math is a powerful tool to understand and describe how things change. Math is a powerful tool to
Everyone can learn the language of algebra. Everyone can learn the language of algebra. Everyone can learn the language
of algebra. Everyone can learn the language of algebra. Everyone can learn the language of algebra. Everyone can learn
You can use patterns to compare relationships and help make the best choice. You can use patterns to compare relation-
ships and help make the best choice. You can use patterns to compare relationships and help make the best choice. You
Algebra helps you solve real-life problems quickly. Algebra helps you solve real-life problems quickly. Algebra helps
Patterns can be used to make predictions and arguments. Patterns can be used to make predictions and arguments.
By generalizing relationships you can understand them more deeply. By generalizing relationships you can understand
There are many ways to represent mathematical situations. Which ways work best to tell your story? There are many
Using and analyzing mathematical models helps you solve complicated problems. Using and analyzing mathematical
Math is a powerful tool to understand and describe how things change. Math is a powerful tool to understand and describe
you solve real-life problems quickly. Algebra helps you solve real-life problems quickly. Algebra helps you solve real-
Patterns can be used to make predictions and arguments. Patterns can be used to make predictions and arguments
them more deeply. By generalizing relationships you can understand them more deeply. By generalizing relationships
ways to represent mathematical situations. Which ways work best to tell your story? There are many ways to represent
models helps you solve complicated problems. Using and analyzing mathematical models helps you solve complicated
how things change. Math is a powerful tool to understand and describe how things change. Math is a powerful tool to
Everyone can learn the language of algebra. Everyone can learn the language of algebra. Everyone can learn the language
of algebra. Everyone can learn the language of algebra. Everyone can learn the language of algebra. Everyone can learn
You can use patterns to compare relationships and help make the best choice. You can use patterns to compare relation-
ships and help make the best choice. You can use patterns to compare relationships and help make the best choice. You
Algebra helps you solve real-life problems quickly. Algebra helps you solve real-life problems quickly. Algebra helps
Patterns can be used to make predictions and arguments. Patterns can be used to make predictions and arguments.
By generalizing relationships you can understand them more deeply. By generalizing relationships you can understand them
MATHteaching in DYS schools
100
you solve real-life problems quickly. Algebra helps you solve real-life problems quickly. Algebra helps you solve real-
Patterns can be used to make predictions and arguments. Patterns can be used to make predictions and arguments
more deeply. By generalizing relationships you can understand them more deeply. By generalizing relationships
101
PATTERNS, RELATIONS, & ALGEBRA
A.
SEQUENCING INSTRUCTION IN DYS
Each strand of the Math Curriculum Framework
is taught during the same three-month period
of time each year.
B.
Each strand of the Math Curriculum Framework
is broken down into discrete topics that address
all of the Emphasized Standards for that strand.
.
PATTERNS, TABLES,
AND RULES
C.
Data Analysis, Statistics and Probability
Patterns, Relations, and Algebra
Geometry and Measurement
Following the calendar ensures that (1) all crucial math information will be addressed, and (2) students
transitioning between DYS settings will be exposed to all elements of a unified math curriculum
LINEAR MODELS
(and a few that
are non-linear)
SYMBOL AND
STRUCTURE
NON-LINEAR
EQUATIONS
PULLING IT ALL
TOGETHER
In treatment facilities), teachers should plan to spend between one and two weeks teaching a MiniUnit on each topic. After proceeding through all topics in the strand, they should then cycle through
the topics again, using different materials, lessons, and examples. By way of contrast, teachers in
very short-term settings should proceed through all strands by spending just one day addressing
each topic, and then cycling through the topics again and again from December through March.
Teachers in all settings must plan carefully to integrate number sense topics and all other topics in
the Patterns, Relations, and Algebra strand.
To teach each strand properly, teachers must
focus on every topic in the strand. The type of
setting and the stability or mobility of students
determines how teachers proceed through all
topics (depicted visually on the following pages).
MATHteaching in DYS schools
September through November
December through March
April through June
102
103
PATTERNS, RELATIONS, & ALGEBRA
SEQUENCING INSTRUCTION IN DYS
ADDRESSING ALL TOPICS IN
DIFFERENT DYS SETTINGS
In classes with very high mobility,
address each topic through Problems
of the Day. Examples are provided
later in this section.
ASSESSMENT SETTINGS
In most settings and classrooms,
instruction should proceed straight
through all topics in the strand.
TREATMENT or OTHER
LONGER-TERM SETTINGS
In treatment settings, instruction
should cycle through all topics, then
repeat in order, incorporating different lessons and examples each time.
ALL DYS SETTINGS
In all settings, Problems of the Day
can be used to enliven a longer unit
of study, signal a change from one
unit to another, or enable students to
apply, deepen, or connect their new
knowledge to what they already
know, understand, and can do.
MATHteaching in DYS schools
ng,
m
r
e
t
t
r
o
Sh
l e ms o
LINEAR MODELS
(and a few that
are non-linear)
PATTERNS, TABLES,
AND RULES
In
t
rea
may be appropriate in setting
”Problems of the D
ay”
SYMBOL AND
STRUCTURE
s with ve
ry high
mobil
i ty
PULLING IT ALL
TOGETHER
NON-LINEAR
EQUATIONS
m
xa
e
d
s an
n
o
less
new
h
t
i
ew
he cycl
t
t
a
e
p
e
r
n
each topic, the
pl
es
DETENTION SETTINGS
“ Prob
l as s l o
ust one c
j
”
,
y
a
f th e D
INTEGRATE
NUMBER SENSE AND
OPERATIONS
INTO ALL
MATHEMATICS
TOPICS, UNITS
& STRANDS
tm
e nt
sett
ings,
procee
d with
104
105
PATTERNS, RELATIONS, & ALGEBRA
BACKGROUND/RESOURCES FOR TEACHERS
These resources provide background knowledge for teachers. They are not student materials, but may help DYS teachers get
PRIMARY RESOURCES
GETTING READY TO TEACH THIS STRAND
ready to teach a unit on Patterns, Relations, and Algebra more effectively and comfortably
WHAT THE RESOURCE OFFERS TEACHERS PREPARING TO TEACH THIS STRAND
CHAPTERS AND/OR PAGES
Chapter 3:
NCTM (National Council of Teachers of Mathematics) Standards for School Mathematics in Algebra
Principles and Standards for School Mathematics
Chapter 7: Data Analysis and Probability Standards
for Grades 9-12 Algebra
Overview of the big algebraic ideas and skills that students should develop Examples of what these ideas look
like in the classroom
Designing and Implementing Mathematics Instruction
Chapter 13, TeachingAlgebraic Thinking, pages 432-484
for Students with Diverse Learning Needs
Overview of how algebraic thinking develops from grades 2-8. Many DYS students
have missed opportunities for basic concepts to solidify, and may therefore benefit from returning to
earlier stages of concept development
EMPower: Seeking Patterns, Building Rules
Unit Introduction, pages xxi - xxvi
An overview of the major themes of algebraic thinking developed throughout the unit
Fostering Algebraic Thinking
Guide for Teachers Grades 6-10 (Driscoll)
Describes algebraic “habits of thinking,” and guides teachers to recognize, develop, and assess these habits in
students Includes algebraic problems to be used in the classroom
Authentic Learning Activities in Middle School
Mathematics
Pages 7-15
Explanation of the NCTM Principles and Standards for School Mathematics Discussion of what’s new in
teaching Patterns, Functions, and Algebra
ONLINE RESOURCES
URL
WHAT THE WEBSITE OFFERS TEACHERS PREPARING TO TEACH THIS STRAND
http://illuminations.nctm.org/ActivitySearch.aspx
Teachers can search for interactive computer activities by topic and grade level
“Illuminations” on the NCTM website
http://illuminations.nctm.org/Lessons.aspx
Shodor Math
http://www.shodor.org/interactivate/activities/index.
html#fun
Science and Numeracy Special Collection
http://www.literacynet.org/sciencelincs/
MATHteaching in DYS schools
Teachers can search for lesson plans and activities by strand, topic, and grade level
Interactive math applets for students to learn algebraic concepts Includes notes for teachers about how and
why to use the applets
Click on “numeracy” Links to many other good websites with recommended resources for teachers and students
106
107
PATTERNS, RELATIONS, & ALGEBRA
TEACHERS ENTER OTHER RESOURCES HERE
ADDITIONAL RESOURCES
PRIMARY RESOURCES
CHAPTERS AND/OR PAGES
NOTES ON THIS RESOURCE
ONLINE RESOURCES
URL
NOTES ON THIS RESOURCE
MATHteaching in DYS schools
108
109
PATTERNS, RELATIONS, & ALGEBRA
CURRICULUM RESOURCES FOR TEACHING
LESSONS AND TOPICS IN THE PATTERNS, RELATIONS, AND ALGEBRA
PRIMARY
CURRICULUM
RESOURCES
PATTERNS, TABLES,
AND RULES
THE TOPICS IN THIS STRAND(continued on next page)
STRAND PROCEED DEVELOPMENTALLY DECEMBER–MARCH
LINEAR MODELS
(and a few that are non-linear)
SYMBOL AND STRUCTURE
NON-LINEAR EQUATIONS
PULLING IT ALL
TOGETHER
Lessons 3-5, Body at Work—
Opening the Unit, pages 1-5
EMPower: Seeking
Patterns, Building Rules
Lesson 1, Guess my Rule,
pages 7-20
Lesson 2, Banquet Tables,
pages 21-32
Bridges to Algebra and
Geometry: Mathematics
in Context
Discovering Algebra: An
Investigative Approach
(2002)
MATHteaching in DYS schools
Tables and Rules, pages 33-49
Graphing the Information, pages 51-64
Pushing it to the Max, pages 65-75
Lesson 11: Rising Gas Prices,
pages 135-146
Lesson 12: The Patio Project,
pages 147-150
Lesson 6: Circle Patterns, pages 77-89
Lesson 7: What Is the Message? pages 91-99
Lesson 8: Job Offers, pages 101-111
Lesson 9: Phone Plans, pages 113-123
Lesson 10: Signs of Change, pages 125-133
Lesson 8.2-8.4
Closing the Unit: Putting It All
Together, pages 151-153
Lessons 4.1 – 4.6,
Activity 2, pages 226-228
Activity 1, pages 449-450
Activity 3, pages 453-455
Lessons and Activity Days
4.4: Linear Plots, pages 206-211
4.6: Linear Equations and Intercept Form, pages 216-224
4.7: Linear Equations and Rate of Change, pages 225-232
Activity Day, pages 242-243
5.1: A Formula for Slope, pages 251-260
5.2: Writing a Linear Equation to Fit Data, pages 261-269
5.3: Point-Slope Form of Linear Equation, pages 270-275
5.5: Writing Point-Slope Equations… pages 284-287
5.6: More on Modeling, pages 288-295
Activity Day, pages 301-302
110
Lesson 4.8: Solving Equations Using the
Balancing Method, pages 233-241
Lesson 5.4: Equivalent Algebraic Expressions,
pages 276-283
111
Lesson 7.2: Exponential
Equations, pages 374-381
Lesson 8.3: Graphs of Real-World
Situations, pages 440-445
Lesson 10.1: Solving Quadratic
Equations, pages 532-537
PATTERNS, RELATIONS, & ALGEBRA
CURRICULUM RESOURCES FOR TEACHING
LESSONS AND TOPICS IN THE PATTERNS, RELATIONS, AND ALGEBRA
PRIMARY
CURRICULUM
RESOURCES
Authentic Learning
Activities in Middle
School Mathematics
PATTERNS, TABLES,
AND RULES
LINEAR MODELS
(and a few that are non-linear)
Warm Up, pages 16-17
Unit 1, Activities 1-4
pages 22-54
Unit 2, Activities 1-3, pages 60-83
How Many Dots…
An Ancient Pattern
From Patterns to Algebra
Formula for the Sum of
Integers from 1 to n
THE TOPICS IN THIS STRAND(continued from previous)
STRAND PROCEED DEVELOPMENTALLY DECEMBER–MARCH
SYMBOL AND STRUCTURE
PULLING IT ALL
TOGETHER
Unit 2 Activity 4:
Making a Choice, pages 84-91
Where Should They Hold the Fundraising Party?
Comparing Galaxy Inn with Noble Pines Country Club
Comparing Holiday Lodge with Noble Pines Country Club
Lessons 1 through 26
Hands-On Equations
5: Coordinates, Slope and Distance
Points Lining Up in the Plane, pages 165-170
The Slope of a Line, pages 171-174
Exploring Algebra with
Geometer’s Sketchpad
The Slope Game pages 175-176
Number Power
Algebra
How Slope is Measured pages 179-182
Pages 92-93, 103-109
Pages 38-91
Number Sense & Operations into instruction in the Patterns Strand
See page 119 for curriculum resources to integrate
MATHteaching in DYS schools
3: Algebraic Expressions, pages 91-104
Equivalent Expressions
Border Problem
Distributive Property
4: Solving Equations & Inequalities, pages 123-140
Approximating Solutions
Undoing Operations
Solving Linear Equations by Balancing
Solving Linear Equations by Undoing
More Slope Games pages 177-178
The Slope-Intercept Form of a Line, pages 201-204
EMPower
Number Power 2
NON-LINEAR EQUATIONS
112
113
PATTERNS, RELATIONS, & ALGEBRA
TEACHERS ENTER OTHER RESOURCES HERE
TEACHERSELECTED
RESOURCES
LESSONS AND TOPICS IN THE PATTERNS, RELATIONS, AND ALGEBRA
PATTERNS, TABLES,
AND RULES
ADDITIONAL RESOURCES
STRAND PROCEED DEVELOPMENTALLY DECEMBER–MARCH
LINEAR MODELS
(and a few that are non-linear)
SYMBOL AND STRUCTURE
NON-LINEAR EQUATIONS
PULLING IT ALL
TOGETHER
Resources for integrating
Number Sense &
Operations into this
strand
MATHteaching in DYS schools
114
115
PATTERNS, RELATIONS, & ALGEBRA
SHORT-TERM PROBLEMS OF THE DAY
CURRICULUM RESOURCES
THESE SHORT 1-DAY MODULES CAN BE DIFFERENTIATED FOR STUDENTS’ VARIED STRENGTHS
PATTERNS, TABLES, AND RULES
“The Pattern Game”
facilitates practice with In and Out tables;
see Math Stories for Problem-Solving Success,
pages 106-109
“Table for Eight?”
lets students role-play a prom committee
trying to determine seating arrangements for
different numbers of attendees;
see Math Stories for Problem-Solving Success,
pages 173-175
“Kids with Stuff”
describes a number of young people with
varying numbers of things; the clues help the
group figure out who has how many;
see Get It Together, pages 28-33
“Number Patterns”
presents problems with clues to help
determine a sequence of numbers;
see Get It Together, pages 132-137
MATHteaching in DYS schools
SHORT MODULES MAY BE PARTICULARLY USEFUL IN DETENTION PROGRAMS
LINEAR MODELS
(and a few that are non-linear)
SYMBOL AND STRUCTURE
Dynamic Geometric Software Activities
give students an introduction to the
relationship between a pattern on the
coordinate system and the equation;
see Exploring Algebra with
Geometer’s Sketchpad, pages 165-168
NON-LINEAR EQUATIONS
“Number Shapes”
uses shapes to represent numbers and provides
clues to solve simultaneous equations using
manipulatives; see Get It Together, pages 34-39
Dynamic Geometric Software Activities
help students connect an intuitive understanding of
slope to its arithmetic calculation;
see Exploring Algebra with
Geometer’s Sketchpad, pages 179-182
“Investigation of Beam Strength”
organizes a lesson where students analyze a
linear relationship between the number of
spaghetti strands used to make a bridge and
the amount of pennies the bridge can support;
see Discovering Algebra, pages 226-227
“Coin Balance”
asks students to use logical reasoning to figure out
how many coins are on a balance; see Authentic
Learning Activities in Middle School Mathematics:
Patterns, Functions, and Algebra, page 18
“Number Tricks”
presents a series of problems that use algebraic
representations to investigate “number tricks”; see
Family Math: The Middle School Years, pages 26-35
116
117
“Investigation into Radioactive Decay”
helps students to create and analyze a model
of radioactive decay; see
Discovering Algebra: An Investigative
Approach (2002), pages 373-374
Dynamic Geometric Software Activities
enables students to plot the graph of a general
quadratic equation and study the effect
of changing the parameters; see
Exploring Algebra with Geometer’s Sketchpad,
pages 225-227
“Growing Quadratic Shapes”
enables students to “grow” squares and cubes and
analyze the relationships between
side length and area or side length,
surface area, and volume; see
Family Math: The Middle School Years,
pages 83-85
PATTERNS, RELATIONS, & ALGEBRA
INTEGRATING
INTEGRATING
NUMBER SENSE
and
OPERATIONS
into the
PATTERNS, RELATIONS, AND
ALGEBRA STRAND
NUMBER SENSE
to integrate into this strand
Properties of Arithmetic
and Algebra
STANDARDS
8.N.8 and 10.N.1
EMPHASIZED STANDARDS
Order of Operations and
Distributive Property
8.N.8
Demonstrate an understanding of the properties of arithmetic operations on rational numbers.
Use the associative, commutative, and distributive properties; properties of the identity and inverse elements
(e.g., -7 + 7 = 0; 3/4 x 4/3 = 1); and the notion of closure of a subset of the rational numbers under an operation
(e.g., the set of odd integers is closed under multiplication but not under addition)
8.N.10
Estimate and compute with fractions (including simplification of fractions),
integers, decimals, and percents (including those greater than 100 and less than 1)
10.N.1
Identify and use the properties of operations on real numbers, including the associative,
commutative, and distributive properties; the existence of the identity and inverse elements for addition
and multiplication; the existence of nth roots of positive real numbers for any positive integer n; and the inverse
relationship between taking the nth root of and the nth power of a positive real number
10.N.2
Simplify numerical expressions, including those involving positive integer exponents or the absolute value,
e.g., 3(24 – 1) = 45, 4 |3–5| + 6 = 14; apply such simplifications in the solution of problems
10.N.3
Find the approximate value for solutions to problems involving square roots and cube roots without the use
of a calculator, e.g., √32-1 ≈ 2.8
Integrating these NUMBER SENSE AND OPERATIONS standards into the PATTERNS, RELATIONS, AND ALGEBRA
STRAND helps students develop a facility with all operations on integers, fractions, and decimals, all of which are required
in algebra. Students should continue to develop number sense through understanding positive and negative numbers and
the properties (commutativity, associativity,and distributivity) and notation of algebra, including order of operations.Most
importantly, when taught alone, Number Sense and Operations can become unnecessarily dry and repetitive. Incorporating
work on Number Sense and Operations into other areas of mathematics can make these fundamental skills come alive!
MATHteaching in DYS schools
NUMBER SENSE and OPERATIONS TOPICS
118
EMPower:Everyday
Number Sense
Lessons 6, 7, and 8
Positive and Negative Integers
Algebraic Notation, including
Exponents and Roots
STANDARD 8.N.10
STANDARDS
10.N.2 and 10.N.3
Concept of Integers
Exponents
EMPower:Everyday
Number Sense
Lesson 5
EMPower: Seeking Patterns,
Building Rules, Symbol Sense
Practice on pages 16-17,
28-29, 44-45, and 85
Number Power Pre-Algebra:
Positive and Negative Numbers
pages 16-17
Inverse Operations
Integer Arithmetic
Discovering Algebra:
An Investigative Approach
Lesson 4.1, Order of Operations
and the Distributive Property
on pages 182-189
EMPower: Seeking Patterns,
Building Rules, Symbol Sense
Practice on pages 61 and 72
Exploring Algebra with
Geometer’s Sketchpad
pages 35-45
Discovering Algebra:
An Investigative Approach
Lesson 4.2; Writing Expressions
and Undoing Operations
pages 190-198
119
Number Power Algebra
Signed Numbers
pages 10-23
Exploring Algebra with
Geometer’s Sketchpad
pages 3-25
EMPower: Seeking Patterns,
Building Rules, Symbol Sense
Practice on pages 144-145
Number Power Pre-Algebra
Using Exponents
page 144
Number Power Algebra
What is a Power?
pages 24-25
Square Roots
Number Power Pre-Algebra
Finding a Square Root, page 145
Number Power Algebra
What is a Power? pages 24-25
Number Power Algebra
What is a Root? pages 32-33
Number Power Algebra
Finding an Approximate
Square Root, pages 34-35
PATTERNS, RELATIONS, & ALGEBRA
SAMPLE MINI-UNIT FOR THIS STRAND
Look over the standards and suggested topics for Integrating Number Sense and Operations.
Choose one standard that you are ready to teach in your classroom.
What resources would you use to teach this standard through stand-alone lessons?
What kinds of preparation would it require? What kinds of results would you expect?
What resources would you use to teach this standard integrated into Patterns, Relations, and
Algebra?
What kinds of preparation would it require? What kinds of results would you expect?
Keep notes about how you teach Number Sense and Operations. What works well, and what
doesn’t work so well? Reflect on how your approach affected students’ skill acquisition,
their engagement in the classroom, and your ability to differentiate instruction.
PRE-ASSESSMENT
How will you determine students’ readiness for this unit? What data will you collect?
What survey of prerequisite learning (i.e., KWL charts, journal prompts, oral surveys)
will you use)?
RESOURCES
Be sure to consider differences in students’ reading levels, interests, readiness to learn
(prerequisite learning), learning styles (multiple intelligences: audio, visual, kinesthetic, etc.)
and backgrounds when selecting culturally responsive resources for the mini-unit.
STAGES OF LESSONS WITHIN THE MINI-UNIT
Introductory Stimulate student interest in the topic, motivate
students to participate in the project
Instructional Students make meaning of content information
and begin to demonstrate, through ongoing
assessment, what they know and understand
Culminating Usually a final assessment, in which students
demonstrate their level of achievement with
regard to the learning objectives
MATHteaching in DYS schools
120
121
PATTERNS, RELATIONS, & ALGEBRA
PATTERNS, RELATIONS, AND ALGEBRA
Learning objectives in this miniunit are tied to the following:
SAMPLE MINI-UNIT
5-8 days
DESIGNER’S NAME
DESIGNER’S EMAIL
Michelle Allman
STRAND
MINI-UNIT TOPIC
Patterns, Relations, and Algebra
LEARNING OBJECTIVES
KNOW…
8.P.4
Create and use symbolic expressions and relate them to verbal,
tabular, and graphical representations.
[email protected]
8.P.5
Exploring Real-World Linear Relationships
Identify the slope of a line as a measure of its steepness and
as a constant rate of change from its table of values, equation,
or graph. Apply the concept of slope to the solution of
problems.
By the end of this mini-unit, students should:
8.P.6
How to represent and analyze linear relationships using
tables, graphs, words, and equations
Identify the roles of variables within an equation, e.g., y = mx
+ b, expressing y as a function of x with parameters m and b.
How to solve linear equations using a variety of
methods
8.P.7
How to use inequality signs to describe relationships
UNDERSTAND…
Set up and solve linear equations and inequalities with one or
two variables, using algebraic methods, models, and/or graphs.
8.P.10
How linear relationships represent many real-world
situations.
Use tables and graphs to represent and compare linear growth
patterns. In particular, compare rates of change and x- and yintercepts of different linear patterns.
How representing a relationship using a table, graph,
equation, and words helps to understand the
relationship better
10.P.6
Solve equations and inequalities including those involving
absolute value of linear expressions (e.g., |x - 2| > 5) and
apply to the solution of problems.
How the slope and y-intercept are connected to the
real-world situation.
…and therefore be able to
DO
MATHteaching in DYS schools
EMPHASIZED
STANDARDS
How the slope and y-intercept are manifested in a table,
graph, and equation.
10.P.7
Solve everyday problems that can be modeled using linear,
reciprocal, quadratic, or exponential functions. Apply
appropriate tabular, graphical, or symbolic methods to the
solution. Include compound interest, and direct and inverse
variation problems. Use technology when appropriate.
Summarize, represent, compare, and analyze three
different pricing methods to rent a facility; complete
related MCAS problems, and discuss the similarities
and differences of these various linear real-world
situations
122
123
PATTERNS, RELATIONS, & ALGEBRA
MINI-UNIT: EXPLORING REAL-WORLD LINEAR RELATIONSHIPS (continued)
OUTLINE OF LESSONS
PRE-ASSESSMENT
From Discovering Algebra, page 226
RESOURCES
Lesson tasks and activities to support students’ achievement of learning objectives
Students will engage in an activity in Lesson One that
provides them with the opportunity to collect data, create
a table, graph, and equation, calculate slope, and use the
equation to determine values. This activity is sufficiently
rich that students can engage with it at a number of
different levels, and their prior knowledge and
background skills will be demonstrated as they complete
the questions.
This lesson enables students to demonstrate what they
already know about linear equations. By the end of this
activity, it will be possible to determine if students know
how to collect data, make a graph, plot points, determine
slope, use variables, write an equation to represent a
situation, and use a graph or an equation to extrapolate data.
MCAS problems
Number Power: Algebra
Discovering Algebra
MATHteaching in DYS schools
Introductory
Students collect data relating to the number of pennies
needed to break a “bridge” made of different amounts of
uncooked spaghetti. They are asked to make bridges of 1,
2, 3, and up to 6 pieces of spaghetti, and to use a table to
record the number of pennies needed to break each
bridge. The text suggests using a graphing calculator to
plot the collected data and linear equation, but if graphing
calculators (or Excel) are not available, students can use
graph paper and pencils. If students do not yet know how
to determine slope or write an equation, the activity can
be modified so the main objectives are to collect data that
is linear, graph their data, and discuss any patterns they
observe. For example, students will likely be able to
identify that for each additional spaghetti strand added,
there is a constant change in the number of pennies the
bridge can support. This recognition can be used to
transition into discussing slope, or rate of change.
Authentic Learning Activities in Middle School
Mathematics
MATERIALS
LESSON ONE
Calculators
Excel software and/or Graphing Calculators, if available
Graph paper
Uncooked spaghetti
Plastic cup
Pennies
String
This is a series of four consecutive lessons in which
students search for the most economical of three places to
hold a fundraising party. Depending on the level of the
students, these problems can be used to develop an
understanding of the structure of linear relationships and
representations, or they may be used to engage students in
a more symbolic approach to the problems.
124
125
LESSONS 2, 4, and 5
PATTERNS, RELATIONS, & ALGEBRA
MINI-UNIT: EXPLORING REAL-WORLD LINEAR RELATIONSHIPS (continued)
From Authentic Learning Activities in Middle School
Mathematics: Patterns, Functions, and Algebra, pages 58-67
LESSON TWO
Instructional
Students are presented with three different price schemes
from three potential sites for a fundraising party. This
activity explores the pricing method of one site that uses a
“minimum charge.” First, students must use the price plan
to determine the cost for different numbers of guests.
They then complete a table and generalize their method
for filling in a table by creating an equation that
represents the situation. If students are struggling with
this step, it may be helpful to (1) have them explain how
they filled in their table, (2) ask them what they would do
if they had some different numbers of guests, and (3) let
them use this repetition to explain what they would do for
any number of guests. If students struggle to substitute
into their formulas and solve for the missing variable,
allow them to use “guess and check” to find the number
of guests for a given cost, and use the lesson below to
practice solving equations.
LESSON THREE
Instructional (optional)
Launch
Model
Practice
Wrap-Up
MATHteaching in DYS schools
From Authentic Learning Activities in Middle School
Mathematics: Patterns, Functions, and Algebra, pages 68-75
This is the second of four consecutive lessons in which
students search for the most economical of three places to
hold a fundraising party. Students extend their work from
Lesson Two to include a pricing method that has a flat
fee. Again, students have the opportunity to understand
the linear relationship by determining the costs for different numbers of guests, and then determining the number
of guests, based on fixed costs. Students then develop a
table, equation, and graph to represent the pricing of this
facility, and compare it to that of Lesson Two.
From Authentic Learning Activities in Middle School
Mathematics: Patterns, Functions, and Algebra, pages 76-83
From Number Power Algebra: Multistep Equations,
pages 68-71. This lesson can reinforce or re-teach how to
solve multi-step linear equations.
LESSON THREE
Instructional
LESSON FOUR
Instructional
This is the third of four consecutive lessons in which
students search for the most economical of three places to
hold a fundraising party. Students extend their work from
Lesson Three to include a new pricing method, and again
have the opportunity to understand the linear relationship
by determining costs for different numbers of guests, and
the number of guests that can be accommodated for
different costs. Students then develop a table, equation,
and graph to represent the pricing of this facility and
compare it to the options presented by the other two sites.
Review the questions in the previous lesson. If students
were using “guess and check” methods rather than
equations, discuss how substituting the known value into
the equation, and then solving for the unknown, can be a
simpler way to solve this type of problem.
Using some of the examples from Number Power, explain
to students the steps to solve a linear equation. Be sure to
remind them that what they are doing is finding the value
that “works” or makes the equation true.
From Authentic Learning Activities in Middle School
Mathematics:Patterns, Functions, and Algebra, pages 84-92
Have students complete a problem set.
This is the last of four consecutive lessons in which
students search for the most economical of three places to
hold a fundraising party.Students will summarize their
work from the previous three classes, analyze the graphs
they have generated, make a recommendation about
which location the committee should choose, and support
their recommendations with data.
Have students share what they learned. Ask them how
they could now solve one of the problems from Lesson
Two, using a method other than guess and check.
126
127
LESSON FIVE
Culminating
PATTERNS, RELATIONS, & ALGEBRA
“PULLING IT ALL TOGETHER”
MINI-UNIT: EXPLORING REAL-WORLD LINEAR RELATIONSHIPS (continued)
This lesson uses real MCAS problems that relate to the
linear relationships students have been investigating
through the lessons in this mini-unit
USING MCAS RESOURCES EFFECTIVELY (continued from previous page)
YEAR
Choose the year in which test items were administered
LESSON SIX
Culminating
GRADE
Choose the grade level associated with test items
Suggested Choices from real MCAS Questions
SUBJECT AREA/QUESTION CATEGORY
Choose the subject area of the test in which items were administered (i.e., Mathematics)
and the question category (i.e., the strand; in this case, Patterns, Relations, and Algebra)
2005 Mathematics Grade 10
Question 18: Short Answer
2005 Mathematics Grade 10
Question 23: Multiple Choice
2005 Mathematics Grade 10
Question 37: Multiple Choice
QUESTION TYPE
Choose multiple choice, open response, short answer, or writing prompt questions
RECOMMENDED MCAS QUESTIONS BY TOPIC, YEAR, GRADE AND TYPE
http://www.doe.mass.edu/mcas/search/
Have students read the information provided in each MCAS
problem and discuss how these problems are similar to the
work in their previous lessons.
LAUNCH
Give students time to work on the rest of the problems,
either independently or in a small group. For the 2005
Mathematics Grade 10 Question 42: Open Response
question, ask them to write their solutions so someone else
would be able to understand their thinking.
PRACTICE
Have students choose one of the problems for the class to
work on together at the board. Be sure to demonstrate how
they should clearly write their solutions and their work.
Using an overhead if possible, show the exemplars that can be
found online, at least for 1-point, 2-point, and 4-point questions.
MODEL
WRAP-UP
USING MCAS RESOURCES EFFECTIVELY (continued on following page)
Whether teachers are using integrated lessons, mini-units, or Problems of the Day, you can find MCAS released
questions that help students review, practice, and apply their learning in this strand. The URL above page will take
you to the Massachusetts Department of Education’s MCAS Question Search Tool.
PATTERNS,
TABLES, AND RULES
2006 Math Grade 10
2006 Math Grade 10
2006 Math Grade 10
2006 Math Grade 10
2006 Math Grade 10
2005 Math Grade 10
2005 Math Grade 10
Spring Retest Question 12
Spring Retest Question 17
Spring Retest Question 22
Question 1
Question 22
Spring Retest Question 1
Spring Retest Question 9
Multiple Choice
Open Response
Multiple Choice
Multiple Choice
Multiple Choice
Multiple Choice
Multiple Choice
LINEAR MODELS
(and a few that are
non-linear)
2006 Math Grade 10
2006 Math Grade 10
2005 Math Grade 10
2005 Math Grade 10
2005 Math Grade 10
2005 Math Grade 10
2004 Math Grade 10
Question 13
Question 17
Fall Retest Question 11
Question 10
Spring Retest Question 25
Spring Retest Question 20
Fall Retest Question 21
Multiple Choice
Open Response
Multiple Choice
Multiple Choice
Multiple Choice
Open Response
Open Response
SYMBOL
AND STRUCTURE
2006 Math Grade 10
2006 Math Grade 10
2004 Math Grade 10
2004 Math Grade 10
Spring Retest Question 11
Question 33
Question 28
Spring Retest Question 32
Multiple Choice
Multiple Choice
Multiple Choice
Multiple Choice
2005 Math Grade 10 Fall Retest Question 35
2005 Math Grade 10 Question 39
2005 Math Grade 10 Spring Retest Question 31
Multiple Choice
Multiple Choice
Open Response
This tool allows users to search real questions that have been used in previous years’ MCAS exams. Teachers do not
have to register to use the MCAS Question Search Tool. Teachers can search released MCAS questions according to
the criteria listed on the following page ( year, grade, subject area/question category, and question type), which are
available through “pull-down menus” on the website.
MATHteaching in DYS schools
128
NON-LINEAR
EQUATIONS
129
PATTERNS, RELATIONS, & ALGEBRA
CONNECTING MATH TO OUR STUDENTS’ LIVES Sample Lessons and Software
Culturally Situated Design Tools are free interactive
“applets” that allow students and teachers to explore
mathematics through connections with cultural
artifacts from specific times, places, and cultures. The
design tools help teachers and students simulate the
development of these types of artifacts, and integrate
learning about specific math topics into standardsbased curricula.
Each site includes sections on cultural background, a
tutorial, software to be used for instructional
activities, and links to teaching materials including
lesson plans, assessment tools (i.e., pre- and posttests), samples of student work from various
instructional settings, and other supports for teachers
This sample lesson is just one example of the teaching
tools that are organized and generously provided to
the public through the Culturally Situated Design
Tools (CSDT) website. All materials are copyrighted
by Dr. Ron Eglash and the Rensselaer Polytechnic
Institute, and are freely available for educational use.
Graffiti can be used as a culturally responsive
instructional tool to teach a number of important math
topics, including Polar Coordinates, Cartesian
Coordinates. Planning, creating, and developing a
personalized style of graffiti relies on a grid, or
coordinate plane, which may be imagined, sketched
on folded paper, or mapped onto bricks.
When connecting math to our students’ lives, it is
important not to trivialize or simplify either the
mathematics or the cultural connections. The CSDT
website includes a valuable section on
that enables teachers to ground their
teaching in history and culture. This section provides
a wealth of information about graffiti’s birth, history
and evolution, as well as its links with art and science.
The teaching materials related to graffiti also include
explicit statements about respecting the law and
creating graffiti on canvas, paper, or a computer, not
tagging community property.
MATHteaching in DYS schools
The images on the right-hand page show how the
(available
free on this website) provides opportunities for
students and teachers to explore grids and coordinates
by mapping the lines, shapes, and groups that
comprise graffiti.
CONNECTING MATH TO OUR STUDENTS’ LIVES
Culturally Situated Design Tools
Graffiti Grapher familiarizes students with x and y
coordinates and explores positive and negative
numbers in a medium that is culturally familiar to
many young people. Cartesian coordinates are used
to locate the start and finish coordinates of each line,
and Polar Coordinates are used to draw the curves,
shapes, arcs and spirals that are graffiti’s most
distinguishing characteristics.
Math concepts in Graffiti art include:
Using different
“shadow” look;
to give graffiti a
Using proper
to keep objects
correctly sized so that viewers can identify what
they are looking at;and
In addition to using pairs of values on the coordinate
plane system (Cartesian coordinates) to create lines
and monitor location, students may also become
interested in exploring the more advanced mathematics that are embedded in graffiti. For example,
students may wish to show location based on
, and use angles and radii
to explore arcs and spirals and create increasingly
beautiful curves in their graffiti.
http://www.ccd.rpi.edu/Eglash/csdt/subcult/grafitti/index.htm
Using
, quadrants and the
coordinate plane to plan ahead and problem-solve
about where objects will go.
130
131
PATTERNS, RELATIONS, & ALGEBRA
APRIL THROUGH JUNE
Emphasized
Standards
Sequencing
Instruction
Background
Resources
for Teachers
Curriculum
Resources
for Teaching
Problems
of the Day
Curriculum
Resources
Integrating
Number Sense
and Operations
Sample
Mini-Unit
teaching in DYS schools
135
136
140
144
148
150
153
Connecting
Math to our
Students’ Lives
162
Pulling It All
Together
164
GEOMETRY AND MEASUREMENT
GEOMETRY AND MEASUREMENT
WHAT DOES THIS STRAND COVER?
Analyzing characteristics and properties of two- and three-dimensional geometric shapes
and developing mathematical arguments about geometric relationships
ESSENTIAL QUESTIONS:
How does the shape of something affect how it can be used?
What is a unit of measurement, and why do we use standard units?
How are one-, two-, and three-dimensional values related to one another?
How does size affect composition?
Why do we use scale models, and how do we create good scale models?
What is the relationship between angles in different geometric shapes?
How can similarity be used to solve problems?
What is the relationship between area and perimeter in various quadrilaterals?
What is the relationship between surface area and volume in 3-D figures?
How are coordinates and special orientation affected by reflection, translation
and rotation?
Specifying locations and describing spatial relationships with coordinate geometry and other representational systems
Applying transformations and using symmetry to analyze mathematical situations
Using visualization, spatial reasoning, and geometric modeling to solve problems
Understanding measurable attributes of objects and the units, systems, and processes of measurement
Applying appropriate techniques, tools, and formulas to determine measurements
WHAT ARE THE EMPHASIZED STANDARDS?
8.G.1
8.G.2
8.G.3
8.G.4
and
10.G.5
8.M.3
8.M.4
10.G.1
10.G.2
10.G.4
10.G.6
10.M.1
10.M.2
teaching in DYS schools
135
Analyze, apply, and explain the relationship between the number of sides and the sums of the interior and
exterior angle measures of polygons
Classify figures in terms of congruence and similarity; apply these relationships to the solution of problems
Demonstrate an understanding of the relationships of angles formed by intersecting lines, including
parallel lines cut by a transversal
Demonstrate an understanding of the Pythagorean theorem and apply the theorem to the solution
of problems; solve simple triangle problems using the triangle angle sum property and/or the
Pythagorean theorem
Demonstrate an understanding of the concepts and apply formulas and procedures for determining
measures, including those of area and perimeter/circumference of parallelograms, trapezoids, and
circles; given the formulas, determine the surface area and volume of rectangular prisms, cylinders,
and spheres; use technology as appropriate.
Use ratio and proportion (including scale factors) in solving problems, including problems involving
similar plane figures and indirect measurement
Identify figures using properties of sides, angles, and diagonals; identify types of symmetry for these figures
Draw congruent and similar figures using a compass, straightedge, protractor, and other tools such
as computer software. Make conjectures about methods of construction and justify the conjectures
with logical arguments.
Apply congruence and similarity correspondences (e.g., ∆ABC ≅ ∆XYZ) and properties of the
figures to find missing parts of geometric figures, and provide logical justification
Use the properties of special triangles (e.g., isosceles, equilateral, 30-60-90º, 45-45-90º) to solve problems
Calculate perimeter, circumference, and area of common geometric figures such as parallelograms,
trapezoids, circles, and triangles
Given the formula, find the lateral area, surface area, and volume of prisms, pyramids, spheres,
cylinders, and cones, e.g., find the volume of a sphere with a specified surface area
GEOMETRY AND MEASUREMENT
A.
SEQUENCING INSTRUCTION IN DYS
Each strand of the Math Curriculum Framework
is taught during the same three-month period
of time each year.
B.
Each strand of the Math Curriculum Framework
is broken down into discrete topics that address
all of the Emphasized Standards for that strand.
.
SHAPES,
CHARACTERISTICS,
AND ANGLES
C.
To teach each strand properly, teachers must
focus on every topic in the strand. The type of
setting and the stability or mobility of students
determines how teachers proceed through all
topics (depicted visually on the following pages).
MATHteaching in DYS schools
September through November
December through March
April through June
Data Analysis, Statistics and Probability
Patterns, Relations, and Algebra
Geometry and Measurement
Following the calendar ensures that (1) all crucial math information will be addressed, and (2) students
transitioning between DYS settings will be exposed to all elements of a unified math curriculum
MEASURES
(and units)
of 2-D SHAPES
3-D SOLIDS AND
MEASURES
SIMILARITY, SCALE,
& PYTHAGOREAN
THEOREM
PULLING IT ALL
TOGETHER
In treatment facilities, teachers should plan to spend between one and two weeks teaching a MiniUnit on each topic. After proceeding through all topics in the strand, they should then cycle through
the topics again, using different materials, lessons, and examples. By way of contrast, teachers in
very short-term settings should proceed through all strands by spending just one day addressing
each topic, and then cycling through the topics again and again from April through June. Teachers in
all settings must plan carefully to integrate number sense topics and all other topics in the Geometry
and Measurement strand.
136
137
GEOMETRY AND MEASUREMENT
SEQUENCING INSTRUCTION IN DYS
ADDRESSING ALL TOPICS IN
DIFFERENT DYS SETTINGS
In classes with very high mobility,
address each topic through Problems
of the Day. Examples are provided
later in this section
ASSESSMENT SETTINGS
In most settings and classrooms,
instruction should proceed straight
through all topics in the strand.
TREATMENT or OTHER
LONGER-TERM SETTINGS
In treatment settings, instruction
should cycle through all topics, then
repeat in order, incorporating different lessons and examples each time
ALL DYS SETTINGS
In all settings, Problems of the Day
can be used to enliven a longer unit
of study, signal a change from one
unit to another, or enable students to
apply, deepen, or connect their new
knowledge to what they already
know, understand, and can do
MATHteaching in DYS schools
ng,
m
r
e
t
t
r
o
Sh
l e ms o
SHAPES,
CHARACTERISTICS,
AND ANGLES
In
t
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MEASURES
(and units)
of 2-D SHAPES
may be appropriate in setting
”Problems of the D
ay”
3-D SOLIDS AND
MEASURES
s with ve
ry high
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SIMILARITY, SCALE,
& PYTHAGOREAN
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PULLING IT ALL
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DETENTION SETTINGS
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INTEGRATE
NUMBER SENSE AND
OPERATIONS
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138
139
GEOMETRY AND MEASUREMENT
BACKGROUND/RESOURCES FOR TEACHERS
These resources provide background knowledge for teachers. They are not student materials, but may help teachers
GETTING READY TO TEACH THIS STRAND
prepare to teach a unit on Geometry and Measurement more effectively and comfortably
PRIMARY RESOURCES
CHAPTERS AND/OR PAGES
WHAT THE RESOURCE OFFERS TEACHERS PREPARING TO TEACH THIS STRAND
Discovering Geometry, Teacher’s Edition
Pages xxvii-xxxvi
Brief discussions of the textbook’s philosophy of geometry education Includes information on how to evaluate
the level at which students are mathematically reasoning Includes ideas about how to structure units and lessons
Teaching Math to Students with Diverse
Learning Needs
Chapters 11 and 12 (Measurement), and Chapter 14
(Geometry)
EMPower: Over, Around, and Within: Geometry and
Measurement, Teacher’s Book
Pages xxi-xxv
Brief guide to using this text book for a geometry and measurement unit
Bridges to Algebra and Geometry, Teacher’s Edition
Pages T10-T16
Because Bridges to Algebra and Geometry addresses both Algebra and Geometry, the information in this guide
focuses on general learning strategies more than those that are specific to Geometry and Measurement
Authentic Learning Activities in Middle School
Mathematics: Geometry and Spatial Sense
Pages 6-12
Introduction to this textbook includes a rationale for teaching measurement Focuses on types of assessments and
includes a bibliography for further reading
ONLINE RESOURCES
URL
WHAT THE WEBSITE OFFERS TEACHERS PREPARING TO TEACH THIS STRAND
National Library of Virtual Manipulatives
http://nlvm.usu.edu
Excellent collection of java-based applets that can be used with a variety of K-12 math activities Index can be
accessed by grade level and/or mathematical strand
NCTM (National Council of Teachers of Mathematics)
Principles and Standards for School Mathematics
University of Illinois at Urbana Champaign
MATHteaching in DYS schools
http://standards.nctm.org
Useful overview of how to differentiate instruction, particularly for students in very heterogeneous settings
Provides helpful examples as well as guidelines
Describes NCTMs standards with specific sections on each strand Includes content strands as well as “process
strands” that can help math teachers focus on process-oriented goals
List of interactive mathematics resources designed for math education Includes a wide range of mathematical
topics For example, in the geometry section the applets range from basic topics (i.e., filling a 3-D solid with unit
cubes) to more advanced topics (i.e., studying matrices and the unit circle)
http://www.mste.uiuc.edu/resources.php
140
141
GEOMETRY AND MEASUREMENT
TEACHERS ENTER OTHER RESOURCES HERE
ADDITIONAL RESOURCES
PRIMARY RESOURCES
CHAPTERS AND/OR PAGES
NOTES ON THIS RESOURCE
ONLINE RESOURCES
URL
NOTES ON THIS RESOURCE
MATHteaching in DYS schools
142
143
GEOMETRY AND MEASUREMENT
CURRICULUM RESOURCES FOR TEACHING
PRIMARY
CURRICULUM
RESOURCES
EMPower: Over, Around, and
Within; Geometry and Measurement
Bridges to Algebra and
Geometry: Mathematics
in Context
Discovering Geometry: An
Investigative Approach
LESSONS AND TOPICS IN THE GEOMETRY AND MEASUREMENT
THE TOPICS IN THIS STRAND
STRAND PROCEED DEVELOPMENTALLY FROM APRIL-JUNE
SHAPES,
CHARACTERISTICS, AND ANGLES
MEASURES
(and units)
of 2-D SHAPES
3-D SOLIDS AND MEASURES
SIMILARITY, SCALE,
& PYTHAGOREAN THEOREM
PULLING IT ALL
TOGETHER
Opening through Lesson 3
Lessons 4 through 7
Lessons 11 through 13
Lessons 4, 9, and 10
Closing Unit
9.1 through 9.4; especially see 9.3
Activity on page 474, and Math Lab
Activity 1 on pages 513-514
11.5 and 11.7
12.1 through 12.5, see especially 12.1,
Activity 2 on page 646)
10.6, 11.2, 11.5, 11.7, and 12.6
see especially 10.6 Activity
on pages 557-558
1.1-1.7, 2.5-2.6, 4.1-4.2,
5.1, and 6.1
6.5. through 6.7 and
8.1 through 8.6
8.7 and 10.1-10.7
4.6, 9.1-9.5, and 11.1-11.6
A unit from either Measurement and
Authentic Learning Activities or
Geometry and Spatial Sense could
be a capstone project
Authentic Learning Activities
in Middle School Mathematics
Pages 116-137 and 138-139
(teacher’s resource book)
Items 212-254
(student book)
Geometry to Go: A
Mathematics Handbook
pages 28-42, 46-47, and 74-87
(teacher’s resource book)
Items 047-081 and 127-167
(student book)
Pages 95-105 and 114-115
(teacher’s resource book)
Items 168-193 (student book)
Pages 106-11
(teacher’s resource book)
Items 194-211 and 320-338
(student book)
Contemporary’s Number
Power 4: Geometry
Angles on pages 9-37
Triangles on pages 38-51
Pages 73-124; application on pages
160-161, 168-171, and 178
Pages 125-145;
applications on pages
166-167, 172-175, and 179
Section 2: Lengths and Angles
pages 38-43
Section 1: Measurements pages 8-17
Section 2: Lengths and Angles
pages 18-37 and 44-47
Pages 52-72
(only covers triangles)
Applications on pages 162-163
and 180-181
Section 4: Capacity and Volume
pages 88-125
Application on page 48
Jamestown’s Number Power:
Measurement
EMPower
Number Power 2
MATHteaching in DYS schools
For an emphasis on proof:
2.1-2.3 and 13.1-13.7
See page 151 for curriculum resources to integrate
144
Number Sense into the Geometry & Measurement strand
145
GEOMETRY AND MEASUREMENT
TEACHERS ENTER OTHER RESOURCES HERE
TEACHER-SELECTED
RESOURCES
LESSONS AND TOPICS IN THE PATTERNS, RELATIONS, AND ALGEBRA
SHAPES,
CHARACTERISTICS, AND ANGLES
MEASURES
(and units)
of 2-D SHAPES
ADDITIONAL RESOURCES
STRAND PROCEED DEVELOPMENTALLY APRIL-JUNE
3-D SOLIDS AND MEASURES
SIMILARITY, SCALE,
& PYTHAGOREAN THEOREM
PULLING IT ALL
TOGETHER
Resources for integrating
Number Sense & Operations
into this strand
MATHteaching in DYS schools
146
147
GEOMETRY AND MEASUREMENT
SHORT-TERM PROBLEMS OF THE DAY
CURRICULUM RESOURCES
THESE SHORT 1-DAY MODULES CAN BE DIFFERENTIATED FOR STUDENTS’ VARIED STRENGTHS
SHAPES, CHARACTERISTICS, AND ANGLES
Origami Angles & Shapes
guides students to make an origami bird while
noting and classifying the shapes and angles
created in each step. An accompanying worksheet
cues students who may need to be refreshed on the
difference between acute and obtuse angles. See
Geometry to Go: Teacher Resource Book,
pages 46-47, and find additional designs at:
http://www.origami-usa.org/fold_this.html
Hands-on Triangle Activities
enable students to work in groups to make as
many triangles as possible while exploring
the angle-sum property. Instructors expand
explorations to include another property of triangles,
(i.e., a triangle must have at least two acute angles)
while students try to find other true statements about
triangles. See Discovering Geometry, Lesson 4.1,
Investigation on page 199, and Bridges to Algebra
and Geometry, Lesson 9.3, Activity on page 474
Hands-on Polygons
follows the previous activity with lessons that
explore the sums of interior angles of polygons.
Students work together to create a large sample of
polygons and compare their findings . See
Discovering Geometry, Lesson 5.1,
Investigation on page 256, and Bridges to Algebra
and Geometry, Lesson 9.4 on pages 481-484 (the
Discovering Geometry lesson, which requires
protractors, enables students to do more of the
discovering for themselves)
MATHteaching in DYS schools
SHORT MODULES MAY BE PARTICULARLY USEFUL IN DETENTION PROGRAMS
MEASURES (and units)
of 2-D SHAPES
3-D SOLIDS AND MEASURES
Rolling Cycloids
helps students explore what happens to a point on a rolling
circle. While this activity is written to be used with
Geometer’s Sketchpad, teachers could, instead, create a large
wheel subdivided into 12 sections, which students roll,
section-by-section along a wall covered with butcher paper or
along the ledge of a chalkboard or whiteboard. Students then
record and measure the position of a certain point fixed along
the edge of the circle. See Discovering Geometry, Exploration:
Cycloids, on pages 346-348
Pick’s Formula for Area
explores an alternative method for finding area, including
the area of irregular shapes. This works especially well
if students also use the MCAS formula sheet method
for finding area, and teachers ask reflection questions
that guide students to compare and contrast methods.
See Discovering Geometry, pages 430-432
More Formulas for Finding Area
explores additional formulas for finding area.
While it is written to be used with Geometer’s Sketchpad,
tasks can also be accomplished with Excel or a scientific/
graphing calculator, and the Visual Thinking activity can use
patty paper or another form of tracing paper.
See Discovering Geometry, Exploration of Alternative
Area Formulas on pages 453-454
“Tiling a Room”
an applied area problem, this activity could be enhanced with
tile props and sample scale rooms (not necessarily rectangular)
to reinforce the 2-dimensional nature of area, and to enable
students to compare different methods of finding area. See
Number Sense: Geometry, pages 164-165
148
SIMILARITY, SCALE,
& PYTHAGOREAN THEOREM
“Scale Down”
works with scale drawings, first using blank paper
Building Solids
and then proceeding to graph paper. See
helps students build shapes and record data to discover
Over, Around, and Within: Geometry and
the relationship between the numbers of edges, faces, Measurement, Lesson 10, Activities 1 and 2 on pages
and vertices in a solid (requires modeling clay and
106-108 of the student book, with ideas on presenting
dried peas, or gumdrops and toothpicks). See
this lesson on pages 109-116 of the teacher’s guide
Discovering Geometry, Exploration: Euler’s Formula
for Polyhedrons, on pages 512-513
“Similar Solids”
explores what happens to the volume and surface area
of similar solids. To adapt for more student
exploration, teachers can bring in paper snow-cone
“Which Can is Most Economical?”
cups and cut off parts of them to create similar cones;
enables students to calculate the surface area of a can,
students can then use water or sand to find the
using cans or other cylindrical props, and paper that
volume of several different similar cones with
can be cut out to cover different surfaces of the cans.
different heights, and then graphing the relationships
See Authentic Learning Activities in Middle School
that result.See Bridges to Algebra and Geometry,
Mathematics: Measurement, Unit 2, Activity 3 on
Lesson 12.6, on pages 680-686
pages 76-83. Differentiated instruction for more
advanced students could couple this lesson with
The Theorem of Pythagoras
Discovering Geometry Lesson 8.7 on pages 445-452, helps students create visual proofs of the Pythagorean
which guides similar investigations into the surface
Theorem by taking squares made on the edges of the
areas of pyramids and cones
legs of a right triangle, and cutting and arranging them
to fit inside a square formed off the hypotenuse of the
triangle. Repeating this activity with several nonsimilar right triangles helps students understand that
“Packaging Softballs”
the Pythagorean Theorem works in all cases. See
guides students through designing a rectangular and
cylindrical box to hold a softball and using calculations Discovering Geometry, Lesson 9.1, on pages 462-427.
to determine which design is more economical
If computers are available, students can do
(requires a softball, two sheets of poster board,
similar activities with java applets at the
scissors, and tape). See Bridges to Algebra and
National Library of Virtual Manipulatives
Geometry, Math Lab Activity 1, on pages 687-689
(http://nlvm.usu.edu/en/nav/frames_asid_164_g_3_t_3
.html?open=instructions)
149
GEOMETRY AND MEASUREMENT
INTEGRATING
INTEGRATING
NUMBER SENSE
and
OPERATIONS
into the
GEOMETRY AND
MEASUREMENT STRAND
NUMBER SENSE
NUMBER SENSE and OPERATIONS TOPICS
to integrate into this strand
Fractions and Decimals
STANDARD 8.N.10
EMPHASIZED STANDARDS
8.N.3
Resources
Use ratios and proportions in the solution of problems, in particular, problems
involving unit rates, scale factors, and rate of change
Number Power 2: Fractions
pages 18-58
8.N.10
Number Power 2: Decimals
pages 60-94
Estimate and compute with fractions (including simplification of fractions),
integers, decimals, and percents (including those greater than 100 and less than 1)
8.N.12

Exponents and Roots
STANDARDS
8.N.12 and 10.N.3
Resources
Number Power: Algebra
pages 24-25, 32-34, and 45
Ratio and Proportion
STANDARD 8.N.3
Resources
EMPower: Keeping Things In
Proportion
Lessons 5 and 6, pages 144-145
Discovering Algebra
Lessons 2.1, 2.2, and 2.3
Select and use appropriate operations—addition, subtraction, multiplication, division, and positive
integer exponents—to solve problems with rational numbers (including negatives)
10.N.3
Find the approximate value for solutions to problems involving square roots and cube roots without the use
of a calculator, e.g., √32-1 ≈ 2.8
When integrating NUMBER SENSE AND OPERATIONS into the MEASUREMENT AND GEOMETRY STRAND
(particularly in Measurement), students should be encouraged to estimate values and use the estimates to check the
reasonableness of answers, employing an understanding of the relative values of and operations on whole numbers,
fractions, and decimals. A knowledge of exponents and roots undergirds interpretation the various units of measure
(i.e., inches vs. inches2 vs. inches3), as well as interpreting geometric formulas (the Pythagorean theorem). When similarity
is discussed, proportionality should be emphasized. Incorporating work on Number Sense and Operations into other areas
of mathematics can make these fundamental skills come alive!
MATHteaching in DYS schools
150
Note that integrating Number Sense and Operations into the Geometry and Measurement strand
requires that teachers repeat (and thereby reinforce) some material
explored previously in the Data and Algebra strands
151
GEOMETRY AND MEASUREMENT
SAMPLE MINI-UNIT FOR THIS STRAND
PRE-ASSESSMENT
Look over the standards and suggested topics for Integrating Number Sense and Operations.
Choose one standard that you are ready to teach in your classroom.
How will you determine students’ readiness for this unit? What data will
you collect? What survey of prerequisite learning (i.e., KWL charts,
journal prompts, oral surveys) will you use?
What resources would you use to teach this standard through stand-alone lessons?
What kinds of preparation would it require? What kinds of results would you expect?
RESOURCES
Be sure to consider differences in students’ reading levels,
interests, readiness to learn (prerequisite learning), learning
styles (multiple intelligences: audio, visual, kinesthetic, etc.),
and backgrounds when selecting culturally responsive resources
for the mini-unit.
What resources would you use to teach this standard integrated into Geometry and
Measurement?
What kinds of preparation would it require? What kinds of results would you expect?
Keep notes about how you teach Number Sense and Operations. What works well, and
what doesn’t work so well? Reflect on how your approach affects students’ skill acquisition,
their engagement in the classroom, and your ability to differentiate instruction.
STAGES OF LESSONS WITHIN THE MINI-UNIT
Introductory Stimulate student interest in the topic, motivate
students to participate in the project
Instructional Students make meaning of content information and
begin to demonstrate, through ongoing assessment,
what they know and understand
Culminating Usually a final assessment, in which students
demonstrate their level of achievement with
regard to the learning objectives
153
GEOMETRY AND MEASUREMENT
GEOMETRY AND MEASUREMENT
Learning objectives in this miniunit are tied to the following:
SAMPLE MINI-UNIT
5-8 days
DESIGNER’S NAME
DESIGNER’S EMAIL
Ryan Casey
STRAND
MINI-UNIT TOPIC
Geometry and Measurement
LEARNING OBJECTIVES
KNOW…
8.G.2
Classify figures in terms of congruence and similarity, and
apply these relationships to the solution of problems
[email protected]
10.G.4
Scale Models in Different Dimensions
Apply congruence and similarity correspondences
(e.g., ∆ABC ≅ ∆XYZ) and properties of the figures to
find missing parts of geometric figures, and provide
logical justification
By the end of this mini-unit, students should:
8.M.1
How to recognize whether a figure or object is or
is not to scale.
Select, convert (within the same system of measurement),
and use appropriate units of measurement or scale
How to use similarity of objects to set up ratios and
solve problems.
8.M.4
Use ratio and proportion (including scale factors) in the
solution of problems, including problems involving similar
plane figures and indirect measurement
How area and volume are affected in similar shapes
and solids.
UNDERSTAND…
10.M.3
Why figures are sometimes drawn to scale and
sometimes not drawn to scale.
Relate changes in the measurement of one attribute of an
object to changes in other attributes, e.g., how changing the
radius or height of a cylinder affects its surface area or volume
How to recognize if two figures are similar.
8.N.3
How similarity is the same and different in
one-, two-, and three-dimensional objects
…and therefore be able to
DO
Use ratios and proportions in the solution of problems, in
particular, problems involving unit rates, scale factors, and
rate of change
Create models that are to scale.
Solve problems by drawing figures that are to scale.
Describe the process of noting whether or not an
illustration has been drawn to scale
MATHteaching in DYS schools
EMPHASIZED
STANDARDS
.
154
155
GEOMETRY AND MEASUREMENT
MINI-UNIT: SCALE MODELS IN DIFFERENT DIMENSIONS (continued)
OUTLINE OF LESSONS
PRE-ASSESSMENT
This lesson allows students to explore scale in a onedimensional setting, and enables teachers to evaluate
students’ abilities to measure length, create and solve
ratios, adjust to different scales, and use a physical model
to represent an abstract idea.
RESOURCES
Lesson tasks and activities to support students’ achievement of learning objectives
In the first lesson, students will explore timelines that are
drawn to scale. Because a timeline is one-dimensional
model, the calculations are pretty straight forward, so
students will be able to demonstrate their understanding
of what is meant by “scale.”
Introductory
First, the instructor will present the students with two
timelines demonstrating the same series of events (events
may be historical or could represent a person’s daily
routine). One timeline should be drawn to scale; the other
should lists the events chronologically with equal
spacing, regardless of intervening time intervals. The
teacher then leads a discussion in which students compare
the way the information is shown in each timeline.
EMPower: Over, Around, and Within:
Geometry and Measurement
Discovering Geometry
Bridges to Algebra and Geometry
Teachers should emphasize the mathematical term,
“scale,” and make sure that the students know what it
means. Students should then look in various sources
(history textbooks, natural science textbooks, or news
magazines) to locate timelines, determine whether or not
the timelines are drawn to scale, and explain how they
made their determinations. Teachers can then ask students
what length of time each inch represents on their
timelines, and demonstrate how this could be determined.
Students compare findings with classmates to find
timelines where an inch represents either more or less
time than their own.
Geometry to Go: Teacher’s Resource Book
pages 120-123, 128, and 139
Number Power: Geometry pages 52-59
MCAS & SAT problems
MATERIALS
LESSON ONE
Rulers
Yard sticks or meter sticks
Various texts that include timelines
Poster board
Butcher paper
Mirror
Graph paper
Blocks
As a project, students can look at chronology charts that
compress the history of the universe into a day, month, or
year. (Many websites explore this idea, originally inspired
by Carl Sagan’s “Cosmic Calendar.”) The Universe
Timeline (URL shown immediately below) is a good
source, as it includes a number of events that are fairly
easy to handle for this project.
http://http://janus.astro.umd.edu/astro/times.html
MATHteaching in DYS schools
156
157
GEOMETRY AND MEASUREMENT
MINI-UNIT: SCALE MODELS IN DIFFERENT DIMENSIONS (continued)
LESSON ONE
(continued from previous
page)
LESSON TWO
Instructional
Students then work as a group to:
1. Use ratios to figure out how many years ago on a real
scale each event happened. They can estimate this
roughly by using the 0 A.D. date provided.
2. Create a physical timeline, drawn to scale, that
represents the history of the universe. The timeline
itself should be as large as possible; using butcher
paper along every wall of the room would be ideal.
3. Try to add a historical or personal event of their own
interest onto the timeline; they should discover that
there is actually no room for this level of detail.
LESSON TWO
From Discovering Geometry, Lesson 11.3. Students solve
problems by using the similarity of shapes, making careful
scale drawings of a problem to come up with a good
estimate of the measures of missing values. Make sure
students understand that the drawings in this and other books
are often not drawn to scale.
LESSON THREE
(optional, for practice
if needed)
To summarize the learning, students can create problems like
the ones in the problem sets and write step-by-step guides
for solving such problems.
To summarize their learning, the class can discuss the
differences between scale and not-to-scale models,
reviewing the advantages and disadvantages of using
timelines drawn to scale, and explaining, in their own
words, how they can use ratios to draw objects to scale.
From Over, Around, and Within: Geometry and
Measurement, Lesson 9. Students continue exploring
scale models by creating 2-dimensional scale drawings,
and solve mathematical problems with similar shapes by
setting up and solving ratios.
Instructional
Students solve problems by figuring out how to use triangle
similarity to set up and solve ratios. The first activity asks
students to find the height of a flag pole, using a mirror and
calculations. When students are not able to leave the classroom, the activity can be done simply by placing an X of
colored tape high on a wall.
Students draw a scale diagram of a door on a blank piece
of paper, and then analyze a mathematician’s statement
and draw a scale diagram using graph paper. After
choosing another subject for a scale drawing, students
make some calculations, using the similarity in shape of
an original object and its scale drawing. For an alternative
and more group-oriented project, students work together
to create a scale drawing of their classroom’s floor plan,
including furniture. See Bridges to Algebra and Geometry
(Teacher’s Resource Book, Enrichment Activity for 11.4)
for a simple worksheet asking students to do this, or
create a list of instructions and provide students with
poster board and a checklist of tasks to complete.
As students start solving problems, teachers can encourage
them to make careful scale drawings of the problems, which
they can measure and convert by their scale factor to find the
answer. They can also solve the same problems algebraically,
as demonstrated in Discovering Geometry. Students should
be able to solve the problems using either method.
In summary, students can compare the effectiveness of
solving these types of problems algebraically or by creating
careful scale drawings and measuring the answers. After
listing the advantages and disadvantages of each method,
student can debate the merits of using illustrations that are
not drawn to scale in textbooks and other printed materials.
To summarize, students explain the process involved in
creating a two-dimensional scale drawing, especially
focusing on how to choose an appropriate scale factor.
Students may also discuss real-life examples of useful 2dimensional scale drawings.
MATHteaching in DYS schools
If the students are having trouble setting up and solving
ratios, a lesson could be spent to give students the
opportunity to practice similarity problems. Problem sets
may be developed from Discovering Geometry, Lesson 11.1,
Geometry to Go (Teacher’s Resource Book), pages 120-123,
128, and 139, and Number Power: Geometry, pages 52-59.
158
159
GEOMETRY AND MEASUREMENT
MINI-UNIT: SCALE MODELS IN DIFFERENT DIMENSIONS (continued)
LESSON FOUR
Instructional
MINI-UNIT: SCALE MODELS IN DIFFERENT DIMENSIONS (continued)
From Discovering Geometry, Lesson 11.5. Students
explore how area and volume are affected as dimensions
change in similar figures or objects. In Activity 1,
students to draw two rectangles on a piece of graph paper,
and discover the relationship between the areas of two
similar shapes, comparing areas in their scale drawings to
the areas of actual surfaces to help them develop or
confirm the relationship.The relationship can also be
explored by referring back to their work from Lesson 2.
This lesson uses real MCAS problems that relate to the
relationships of similarity and scale that students have been
investigating through the lessons in this mini-unit
In Activity 2, students explore similar solids. If the
Discovering Geometry blocks are not available, any
blocks can be used, or if no blocks are available, sugar
cubes may be an inexpensive substitute.Some time must
be spent ensuring that students understand what is meant
by similarity in 3-dimensional objects. Teachers should
make sure that students scale in all directions.
Culminating
MATHteaching in DYS schools
Culminating
http://www.doe.mass.edu/mcas/search/
Encourage students to communicate their thought
processes, including illustrating the problems when
pictures are not provided. Students should be able to state,
in their own words, how volume and area are affected as
shapes and solids are scaled, and may also journal about
the process of discovering evidence and making deductions.
LESSON FIVE
LESSON SIX
Students consider strategies for taking standardized tests,
demonstrating their ability to solve questions involving
scale drawings and similarity. After giving students
several similarity problems from the MCAS and/or the
SAT, ask them to consider the claim that objects on these
tests are not necessarily drawn to scale. Have them
examine several figures and determine whether or not the
questions are or are not drawn to scale, while discussing
the pitfalls of assuming that a figure is drawn to scale.
Students should choose one figure that was not drawn to
scale, and re-draw it so that it is drawn to scale.
Students then examine sample solutions to some openresponse MCAS questions involving similarity or scale
drawings, checking their work against a rubric,
comparing it with other students’ work or exemplars, and
then revising their work.
160
Have students read the information provided in each MCAS
problem and discuss how these problems are similar to the
work in their previous lessons.
Launch
Give students time to work on the rest of the problems,
either independently or in a small group.Ask them to write
their solutions so someone else would be able to understand
their thinking.
Practice
Have students choose one of the problems for the class to
work on together at the board. Be sure to demonstrate how
they should clearly write their solutions and their work.
Model
Using an overhead if possible, show the exemplars that can
be found online, at least for 1-point, 2-point, and 4-point
questions.
Wrap-Up
161
GEOMETRY AND MEASUREMENT
CONNECTING MATH TO OUR STUDENTS’ LIVES Sample Lessons and Software
Culturally Situated Design Tools are free interactive
“applets” that allow students and teachers to explore
mathematics through connections with cultural
artifacts from specific times, places, and cultures. The
design tools help teachers and students simulate the
development of these types of artifacts, and integrate
learning about specific math topics into standardsbased curricula.
Each site includes sections on cultural background, a
tutorial, software to be used for instructional
activities, and links to teaching materials including
lesson plans, assessment tools (i.e., pre- and posttests), samples of student work from various
instructional settings, and other supports for teachers
This sample lesson is just one example of the teaching
tools that are organized and generously provided to
the public through the Culturally Situated Design
Tools website. All materials are copyrighted by Dr.
Ron Eglash and the Rensselaer Polytechnic Institute.
can be useful in teaching
a number of math and geometry topics, including
transformational geometry (translation, rotation,
dilation, and reflection), ratio, proportion, angles,
iteration, geometric sequence, Cartesian coordinates,
circles, logarithmic spirals, and exponents.
CONNECTING MATH TO OUR STUDENTS’ LIVES
provides
links to information about the History of Cornrow
Braiding, as well as tutorial on How to Create Braids,
and How to Position Braids.
(see right-hand page) enables
teachers or students to simulate creation of cornrow
curves, which are followed by
Each plait (y shape) in the braid is scaled down
by 90% of the previous plait.
a. If the first plait is 1 inch wide, how wide is the
second? (answer = 0.9 inches)
b How wide is the third?
c. How wide is the nth plait?
For a braid with no scaling (“dilation”), and rotating
by 1 degree in each plait:
a. How many plaits will be needed to make a
complete circle? (answer: 360)
b. How many using a 10 degree rotation?
c. How many for an n degree rotation?
is another good subject to teach through
Cornrow Curves. A tutorial section on dilation
includes sample questions for students on ratio, for
example:
Each plait (y shape) in the braid is scaled down
by 90% of the previous plait.
a. If the first plait is 1 inch wide, how wide is the
second? (answer: 0.9 inches)
b. How wide is the third? (answer: 0.81 inches)
c. How wide is the nth plait? (answer: 0.9n)
http://www.ccd.rpi.edu/Eglash/csdt/african/CORNROW_CURVES/
MATHteaching in DYS schools
Culturally Situated Design Tools
162
Teachers can also look for specific
to explore;
circles and spirals are excellent examples. Geometry of the
circle, for instance, can be explored by looking at the relations
between rotation and iteration. A circular braid will be
generated any time you have a braid with rotation and
sufficient numbers of plaits (that is, sufficiently high number
of iterations). Students might do an inquiry exercise:
It depends on the rotation—the higher the rotation, the fewer
plaits you need. If you are only rotating by 1 degree in each
plait, then you will need 360 plaits to go full circle (that’s only
359 iterations, because you get one plait to start with; in other
words there is a “zeroth” iteration at start). A 10 degree
rotation will require only 36 plaits to make a full circle, and
so on. Having students discover this relationship on their own
can be an empowering exercise.
163
, including pre-and posttests for the Cornrow Curves lessons, are also
available on the website.
GEOMETRY AND MEASUREMENT
“PULLING IT ALL TOGETHER”
“PULLING IT ALL TOGETHER”
USING MCAS RESOURCES EFFECTIVELY
USING MCAS RESOURCES EFFECTIVELY
Whether teachers are using lessons, mini-units, or Problems of the
Day, you can find MCAS released questions that help students
review, practice, and apply their learning in this strand.
LAUNCH
This tool allows users to search real questions that have been used
in previous years’ MCAS exams. Teachers do not have to
register to use the MCAS Question Search Tool.
PRACTICE
MODEL
The URL on the right will take you to the Massachusetts
Department of Education’s MCAS Question Search Tool.
WRAP-UP
Have students choose one of the problems for the class to work on together at the
board. Be sure to demonstrate how they should clearly write their solutions and
their work.
Give students time to work on the rest of the problems, either independently or
in small groups. For Open Response questions, ask students to write their
solutions so that another person would be able to understand their thinking.
Using an overhead if possible, show the exemplars that can be found online,
at least for 1-point, 2-point, and 4-point questions.
RECOMMENDED MCAS QUESTIONS BY TOPIC, YEAR, GRADE AND TYPE
http://www.doe.mass.edu/mcas/search/
SHAPES,
CHARACTERISTICS,
AND ANGLES
Teachers can search released MCAS questions according to
the following criteria, which are available through “pull-down
menus” on this website:
YEAR
Choose the year in which test items were administered
MEASURES
(and units)
OF 2-D SHAPES
GRADE
Choose the grade level associated with test items
SUBJECT AREA/QUESTION CATEGORY
Choose the subject area of the test in which items were administered (i.e., Mathematics)
and by the question category (i.e., the strand; in this case, Geometry and Measurement)
QUESTION TYPE
Choose multiple choice, open response, short answer, or writing prompt questions
SOLIDS AND
MEASURES
Each of the MCAS math questions will also indicate clearly (using a graphic of a calculator, with or without a large red
X) whether students were allowed to use a calculator for this question when taking the MCAS. Lessons that include
MCAS released items should follow the same guidelines.
MATHteaching in DYS schools
Have students read the information provided in each MCAS problem and discuss
how these problems are similar to their work in previous lessons.
164
SIMILARITY, SCALE,
& PYTHAGOREAN
THEOREM
165
2006 Math Grade 10
2006 Math Grade 10
2005 Math Grade 10
2004 Math Grade 10
2004 Math Grade 10
2006 Math Grade 10
2006 Math Grade 10
2005 Math Grade 10
2005 Math Grade 10
2005 Math Grade 10
2005 Math Grade 10
2005 Math Grade 10
Question 9
Question 39
Question 19
Question 10
Question 36
Question 37
Question 41
Question 9
Question 15
Question 21
Question 24
Question 40
Multiple Choice
Multiple Choice
Short Answer
Multiple Choice
Multiple Choice
Multiple Choice
Open Response
Multiple Choice
Short Answer
Open Response
Multiple Choice
Multiple Choice
2006 Math Grade 10, Question 18: Short Answer
2006 Math Grade 10, Question 23: Multiple Choice
2006 Math Grade 10, Question 38: Multiple Choice
2005 Math Grade 10, Question 27: Multiple Choice
2004 Math Grade 10, Question 41: Open Response
2006 Math Grade 10
2006 Math Grade 10
2006 Math Grade 10
2005 Math Grade 10
2004 Math Grade 10
2004 Math Grade 10
Question 26
Question 28
Question 32
Question 32
Question 16
Question 25
Multiple Choice
Multiple Choice
Multiple Choice
Multiple Choice
Short Answer
Multiple Choice
GEOMETRY AND MEASUREMENT
ASSESSMENT
PRINCIPLE (NCTM)
DEFINING ASSESSMENT
BALANCED ASSESSMENT
ONE SIZE DOESNOT FIT ALL
ASKING QUESTIONS WITH
BLOOM’S TAXONOMY
teaching in DYS schools
173
174
175
176
177
ASSESSING PROGRESS
TOWARD MEETING THE
LEARNING OBJECTIVES
178
USING RUBRICS FOR
AUTHENTIC ASSESSMENT
179
LOOKING AT STUDENT
WORK—MCAS AND RUBRICS
180
CREATING AND FINDING
ASSESSMENTS AND
RUBRICS ONLINE
185
ASSESSMENT
DEFINING ASSESSMENT
Assessments include many different methods of gathering evidence to measure student progress in learning crucial material. The various assessment methods used in DYS settings may include:
Assessment should support the learning
of important mathematics, and furnish
useful information to both
teachers and students.
PRE-ASSESSMENT
Prior to beginning a mini-unit of instruction, teachers gauge what students know, understand, and are able
to do. Formal pre-assessments gather data that is specific to each student, while informal pre-assessments rest
on general data for a group of students. All pre-assessments should target the primary learning objectives of the
mini-unit (what students should know, understand, and be able to do by the end of the mini-unit).
Assessment should be more than merely a
test at the end of instruction to gauge
learning. It should be an integral
part of instruction that guides
teachers and enhances
students’ learning.
ASSESSMENT
Teachers observe learning by describing, collecting, recording, scoring, and interpreting information about a
student’s learning. Data may be used to adjust instruction, coach students, or assist in final evaluation of student
progress. Assessment data may or may not be quantitative in nature.
Teachers should be continually
gathering information about their
students through questions, interviews,
writing tasks, and other means.
They can then make appropriate
decisions about such matters as
reviewing material, re-teaching
a difficult concept, or providing
something more or different for
students who are struggling
or need enrichment.
FORMATIVE assessment is conducted before or during instruction to provide teachers with data regarding
the degree to which a student knows, understands, or is able to do a given learning task. Information from
formative assessment is useful in planning and sequencing students’ learning experiences, and can be
particularly useful in coaching students.
SUMMATIVE assessment takes place at the end of an instructional unit, and provides information on
student performance relative to the learning objectives outlined in the mini-unit plan. Information from
summative assessment is used to make a judgment or evaluation of student accomplishments in that
mini-unit, and comprises a critical part of student evaluation.
PERFORMANCE-BASED ASSESSMENT
Teachers observe and assess student performance in projects, presentations, or performances using a set of
established criteria. Because performance-based assessment is essentially subjective, teachers must use a
scoring guide, or “rubric,” that is based on explicit criteria and clear descriptions of various levels of quality.
Assessments should focus on
understanding as well as procedural
skills. Because different students
show what they know and can do
in different ways, assessments should
also be conducted in multiple ways.
Teachers should look for a convergence
of evidence from different sources.
Teachers must ensure that all students
are given an opportunity to demonstrate
their mathematics learning.
HOLISTIC RUBRICS combine a number of elements of performance into a short descriptive narrative for
each scoring level. The emphasis is on evaluating the overall product or performance.
ANALYTIC RUBRICS separate
National Council of Teachers of Mathematics
(NCTM) Principles and Standards Guide
the performance or product into its critical attributes, and each category or
attribute is evaluated separately. Because it provides specific information about the various components of the
performance or product, this type of rubric is most useful as a coaching tool.
PORTFOLIO ASSESSMENT
Teachers evaluate a collection of each student’s work, using a pre-established set of criteria. Because
performance-based assessment is essentially subjective, expectations for content and criteria for assessment
must be clear to students and teachers before portfolios are created or assessed.
PORTFOLIOS include work that is representative of each student’s efforts, achievements, and progress over
a period of time Portfolios may be evaluated by scoring each piece individually, scoring of a set of pieces as a
whole, or simply confirming that each required component has been included. Portfolios may include a wide
range of products that demonstrate student learning, including (for example) videotapes, audio tapes, journals,
completed assignments, quizzes, tests, or other sample work.
MATHteaching in DYS schools
174
175
ASSESSMENT
BALANCED ASSESSMENT—ONE SIZE DOES NOT FIT ALL
ASKING QUESTIONS WITH BLOOM’S TAXONOMY
In DYS settings, balanced assessment means that teachers gather information about students’ learning progress
throughout the instructional process, and in a variety of ways. Recognizing that “one size does not fit all,” teachers
differentiate their assessment approaches to meet the needs of students with diverse learning styles, multiple
intelligence preferences, and other considerations.
On an ongoing basis, teachers’ formative
assessment techniques include asking questions,
(verbally or on worksheets), observing students
during work sessions and activities, creating “ticket
to leave /exit card” activities, giving quizzes, and
assigning journal entries.
KNOWLEDGE
At the end of each mini-unit, teachers gather comprehensive data about students' progress relative to the
learning objectives of the unit .These summative assessment activities may include performance tasks, projects, or
comprehensive tests, and are commonly used to make a final evaluation of student progress for transcripts.
FEATURES OF BALANCED ASSESSMENT
Recognizing the diversity of the student population,
teachers provide flexibility in the assessment
process to allow students to demonstrate their
knowledge and understanding in a
variety of ways.
Learning objectives for the mini-unit
or lesson are clearly communicated
to students; students know what we
want them to Know, Understand,
and be able to Do.
EVALUATION
While Benjamin Bloom’s name was alphabetically
first in a list of experts who developed this
classification, many college and university
professors participated jointly in developing what
is now known as Bloom’s Taxonomy.
In the 1990’s, a former student of Bloom’s named
Lorin Anderson led a team of cognitive psychologists in reviewing and revising the original
taxonomy. To reflect the active nature of thinking,
the name of each category of thinking was changed,
and some categories were renamed to reflect the
quality of these thinking processes.
Classifications in both the original and revised
taxonomies are useful in asking questions and
developing assignments that promote higher-order
thinking. Using this taxonomy helps teachers assess
student progress in ways that are grounded in
different thinking processes.
REMEMBERING
UNDERSTANDING
APPLYING
ANALYZING
EVALUATING
CREATING
Teachers use “prompts” from Bloom’s Taxonomy to assess the level or degree to which students grasp the material. The
following examples illustrate this concept by focusing on the learning objective “To Produce, Use, and Comprehend
Quantitative Information in Real-World Situations.”
Recognize, describe, and name math concepts, facts, and skills related to real-world situations
BL
IN
K
G
177
Explain, compare, & outline appropriate math concepts, facts and skills related to real-world situations
Use math concepts, facts and skills to examine and solve real-world situations mathematically
Distinguish strengths and weaknesses of using particular math concepts, facts and skills to describe
real-world situations, and categorize their different points of view, biases, values, or intents
IN
176
SYNTHESIS
OMI
Student portfolios are used to collect student work
as a form of assessment, with key pieces of work
selected by the students to meet
established criteria for evaluation or to
demonstrate progress.
ANALYSIS
O
MATHteaching in DYS schools
Prompts that involve verbs from
higher-level thinking processes
(outlined in Bloom’s Taxonomy,
see facing page), with an emphasis
on evaluating, creating, applying, and
analyzing, are used for culminating
performances and complex or other
assessment projects.
APPLICATION
TH
Reflective processes and activities include
self-reflection, peer coaching, journals,
logs, and self-critiques.
Tests and quizzes include a variety of response types,
including true/false or multiple choice selections, as
well as responses that students must develop
themselves, such as problems to solve,
short answer, open-response or
performance tasks.
COMPREHENSION
AN
Teachers use a range of assessment
tools to monitor (formative assessment)
and evaluate (summative assessment)
students’ progress.
For more than 50 years, Bloom’s “Taxonomy of Educational Objectives” has been used as a valuable tool to organize
educational goals and promote high-order thinking. The taxonomy classifies six levels of qualitatively different
thinking processes, with different kinds of thinking organized in a clear hierarchy. One end of the classification is
considered basic thinking skills (factual or topical knowledge and retrieval), while the other end comprises
higher-level thinking skills (conceptual understanding needed for critical thinking and problem-solving).
Invent or design products that involve particular math concepts, facts or skills in real-world situations
Recommend and prioritize a number of different solutions to a particular real-world problem,
justifying your assessment.
ASSESSMENT
USING RUBRICS FOR AUTHENTIC ASSESSMENT
ASSESSING PROGRESS TOWARDS THE MEETING THE LEARNING OBJECTIVES
Instructional activities in DYS are focused on concrete Learning Objectives, expressed in terms of what we want
students to know, understand, or be able to do. But what does this mean in concrete terms? How can we discern what
a student knows, understands, and is able to do?
How do we know what a
student knows, understands,
and is able to do?
Rather than jump to conclusions about whether or
not a student has grasped a particular body of
knowledge, it can be helpful first to slow down,
take a deep breath, and simply
what
we see. Describe in detail what you see happening.
You may describe this in your mind, with a colleague, in your journal, or in the privacy of your own home. What is
the student doing? What is she not doing? For example, is your student adding the ones in the tens column and
counting them as units rather than tens? Multiplying the number in the tens column and placing the total in the ones
column? Drawing a picture to show the number of boxes of cereal she can buy for ten dollars? Using his fingers to
multiply by fives?
After noting what is happening,
what that means—what does your student
know or not know? Can your student multiply correctly, but writes the digits in the wrong place because he doesn’t
understand that the one in the tens column represents ten, not one? Can the student add correctly, but doesn’t seem to
understand what the digits in the tens column represent? Is she able to determine the ratio of cereal to dollars? Can he
count by fives, but doesn’t know his 5 facts by memory?
Following a review of your students’ actions and what they suggest about their knowledge, then
you will take to help your students progress. How will you help? What are your next steps? For example,
you might have students practice with concrete manipulatives for example, popsicle sticks), bundling groups of ten
with rubber bands so that they can see that one in the tens place is really one bundle of ten, and five of these bundled
sets total 50, not five. You could have your student continue drawing cereal boxes, then replace the drawings with
numbers and see if she can still recognize the pattern. Or you might ask your student to start looking for patterns in
the five table (i.e., even number answers end in 0, odd numbers end in 5), and practice his 5 facts out of order.
In her work on “computational fluency,” Dr. Susan Jo Russell, a leader at the Education Research Collaborative at
TERC in Cambridge, Massachusetts, outlines precisely what a numerate person (an adult, in this example) must
know, understand, and be able to do, using a simple problem (23 x 13) as an illustration. Although this illustration
is focused explicitly on adult numeracy, it presents wonderfully concrete examples to illuminate the conceptual
framework of knowing, understanding, and being able to do:
Confronted with the problem,
, a numerate adult has the capacity to: Punch the numbers
into a calculator and achieve precise results Use a pencil-and-paper algorithm to achieve precise
results Decompose the numbers mentally to achieve precise results (23 x 10 = 230 + 23 x 3 = 230
+ 69 = 299) Round quantities to estimate an answer (20 x 10 = 200, so the answer must be greater
than 200) Picture 23 on a number line with that interval repeated 13 times Picture an array
with 23 rows and 13 columns Picture 13 groups of 23 objects Envision a situation represented
by the numbers (e.g., I have to make 13 payments of 23 each, so it’s a year plus one month. or
23 x 12 +23) Recognize the inverse situation (so in a division problem, I’d take the total and
divide it by 23 to yield an answer of 13).
MATHteaching in DYS schools
178
refers to methods that correspond as closely as possible to real-world
experiences. These techniques were first applied in arts and apprenticeship systems, where assessment has always
been based on performance.Authentic assessment takes these principles of evaluating real work into all areas of the
curriculum. In using “authentic assessment,” the instructor will:Observe the student in the process of working on
something real Provide feedback Monitor the student’s use of the feedback, and Adjust instruction and
evaluation accordingly.
are particularly useful in assessing student knowledge, skills, or applications on performances
(such as a speech, debate, or PowerPoint presentation) or products (such as a written response, the results
of a project, or a portfolio of work). In DYS settings, teachers use rubrics as scoring guides to evaluate the quality of
responses constructed by students in
and
assessments. Using rubrics, students
can become involved in both peer- and self-assessment. As students become familiar with rubrics, they can also assist
in the process of designing the rubrics. This involvement empowers students and contributes to more focused and
self-directed learning.
Rubrics focus on responses constructed by students in their own performances and products. This emphasis is
quite different from multiple-choice, matching, or similar teacher-constructed choices for responses. The
advantages of using rubrics in assessment are that they allow assessment to be more
and consistent,
focus teachers to clarify their expectations (
) in explicit terms, show students exactly what is
and how their work will be evaluated, promote student
about the criteria to use in
assessing peer performance, provide teachers with useful
regarding the effectiveness of the
instruction; and offer students and teachers benchmarks against which to measure and
All rubrics have three
, described briefly below.
CRITERIA OR STANDARDS:
.
The learning outcomes that the student is demonstrating through the work.
In the DYS mini-unit and lesson framework, these would be the objectives (what students should know,
understand and be able to do) and the corresponding standards from the Massachusetts Curriculum Framework.
Criteria or standards will vary from rubric to rubric, even within the same mini-unit, depending on the learning
that is being assessed.
QUALITY DEFINITIONS:
Describe the way that differences in students’ responses will be judged.
For example, if a particular question requires that students provide a correct numeric answer and demonstrate the
process they used and provide a written explanation, the rubric must indicate which of these components will be
assessed in awarding a score. While there are many possible options for labeling each level, the rubric must
provide a separate description for each qualitative level (i.e., a 1-4 scoring guide).
SCORING STRATEGY:
May be either holistic or analytic.
In a holistic strategy, the scorer takes all of the criteria into consideration but aggregates them to make a single,
overall quality judgment. In an analytic strategy, the scorer gives criterion-by-criterion scores, so every criterion
on a particular product or performance is given a separate score. Most commonly, the scorer gives a rating for
each criterion, and then also gives a total score (usually by adding up the criterion scores).
179
ASSESSMENT
LOOKING AT STUDENT WORK—MCAS ASSESSMENTS AND RUBRICS
LOOKING AT STUDENT WORK—MCAS ASSESSMENTS AND RUBRICS
4 POINTS
To graduate from a Massachusetts high school, students must
demonstrate competency on the MCAS (Massachusetts
Comprehensive Assessment System). In all subject areas,
including mathematics, MCAS assessments rely heavily
on examinations of students’ own work. For this reason,
published MCAS rubrics provide a helpful guide to the
process of looking at student work. The following two
examples, released and published by the Department
of Education, illuminate the process of using
explicit rubrics to assess student work.
EXAMPLE 1 On the 2005 MCAS for Grade 8
Math, Question 28 was based on the Measurement and Geometry strand. Students were
provided with a ruler, a scaled map of a
campground in a state park, and a scale that
showed 1 inch = ½ mile. Students were asked a two-part question with instructions to show or explain how they
got their answers. “Based on the scale, what is the distance, in miles, from the park entrance to Antler Bluff?” and
“What is the area, in square miles, of the campground?”
Open-response questions like this one require students to generate ( rather than recognize) a correct response. Students
can respond correctly using a variety of strategies and approaches, and scoring allows students to receive credit for
different strategies and approaches. MCAS scoring guides indicate what knowledge and skills students must demonstrate
to earn specific numbers of score points (0-4). Answers to these questions are not scored for spelling, punctuation, or
grammar.
Score
Description
4
The student response demonstrates an exemplary understanding of the [name of strand] concepts involved
in [brief description of the skills involved in solving the problem].
3
The student response demonstrates a good understanding of the [name of strand] concepts involved in
[brief description of the skills involved in solving the problem]. Although there is significant evidence that
the student is able to recognize and apply the concepts involved, some aspect of the response is flawed. As a
result, the response merits 3 points.
2
The student response demonstrates a good understanding of the [name of strand] concepts involved in
[brief description of the skills involved in solving the problem]. While some aspects of the task are
completed correctly, others are not. The mixed evidence provided by the student merits 2 points.
1
The student response demonstrates only a minimal understanding of the [name of strand] concepts
involved in [brief description of the skills involved in solving the problem].
0
The student response contains insufficient evidence of an understanding of the [name of strand] concepts
involved in [brief description of the skills involved in solving the problem] to merit any points.
MATHteaching in DYS schools
180
2 POINTS
181
ASSESSMENT
LOOKING AT STUDENT WORK—MCAS ASSESSMENTS AND RUBRICS
EXAMPLE 2 On the 2005 MCAS for Grade 10 Mathematics, Question 17 was based on the Patterns, Relations, and
LOOKING AT STUDENT WORK—MCAS ASSESSMENTS AND RUBRICS
Algebra strand. Students were given the following information about a hypothetical worker named Quinn who works in
Chicago and New York City, and travels by taxi in each of the two cities. In Chicago, he pays a fixed taxi fare of $1.90
per ride, plus $1.60 per mile traveled. In New York City, he pays a fixed taxi fare of $1.50 per ride, plus 25¢ per mile
traveled.
1 POINT
Students were asked to write equations and compute distances as follows:
a. Write an equation that expresses f, Quinn’s total fare for a taxi ride in Chicago, as a function of m,
the number of miles traveled.
b. Write an equation that expresses f, Quinn's total fare for a taxi ride in New York City, as a function of m, the
number of miles traveled.
c. On a recent trip, Quinn noticed that the total number of miles traveled by taxi from the airport to the hotel was
the same in each of the two cities. Before tips were added, his taxi fare to the hotel in New York City was
$12.20 more than his taxi fare to the hotel in Chicago. What was the distance from the airport to the hotel in
each city? Show or explain how you got your answer.
0 POINTS
3 POINTS
The MCAS release items highlighted in this manual share a common rubric that is easily adaptable
for different grade levels, strands, standards, and skills.
MATHteaching in DYS schools
182
183
ASSESSMENT
CREATING AND FINDING ASSESSMENTS AND RUBRICS ONLINE
Think about a particular math skill, lesson, or mini-unit that you have enjoyed teaching.
What assessment methods did you use? What roles did the assessments play in your planning
and instruction?
A great many websites help teachers find and create appropriate rubrics and assessment instruments online.
At the time of publication, all of the following websites were working well, and provided
teachers with assessments and tools to generate and/or customize high-quality rubrics at no cost.
What factors do you consider when selecting an assessment method?
http://rubistar.4teachers.org
Are particular methods of assessment better suited to different types of curriculum and
instruction? Why or why not?
http://www.rubrics4teachers.com
Quickly scan this list of different ways to gather evidence of student learning. Which methods
do you use most? Why? Are there methods that you never use?Why not? What methods would
you like to try for the first time?
http://literacy.kent.edu/Midwest/assessment
http://www.teach-nology.com/web_tools/rubrics
Multiple choice tests or quizzes
Figural representations
Filling in the blanks
Learning logs
Labeling a diagram, map, etc.
Process folios
True/false test or quiz
Demonstrations
Diaries or journals
Competitions (e.g. athletic competition)
Musical, dance, or dramatic performances
Science fairs (or similar demonstrations)
Newspaper advertisements or other media
Web page or other internet products
Portfolios of work
Observations of students
Concept maps
http://www.nwrel.org/assessment/toolkit98.php
Essays, stories, or poems
Matching
Think alouds
http://school.discovery.com/schrockguide/assess.html
http://www.4teAchers.org/projectbased/checklist.shtml
“Show your Work”
Debates
Interviews
Short answers
Oral questioning
Skills tests
Research reports
Oral presentations
185
ASSESSMENT
Acknowledgements
This math guide is the second in a series of instructional guides that focus on
the content and delivery of education services in DYS facilities across the state
of Massachusetts.The DYS Instructional guides are one component of the DYS
LEED Education Initiative, an education reform effort led by the Commonwealth
Corporation, under contract with the Massachusetts Department of Youth
Services. All materials are in these guides are aligned with the Math Curriculum
Framework and the content standards from the Massachusetts Department of
Education.
The guidance and good sense within these pages comes from dozens of talented
and dedicated practitioners who have generously shared their efforts and
expertise.The following individuals and organizations were instrumental in creating the final product:
William Diehl, Janice Manfredi, Michelle Allman, Monique Miles, Anika Nailah,
Talitha Abramsen,Tanya Lieberman, Ryan Casey, and Kathy Rho of the Commonwealth Corporation in Boston, MA
Mary Jane Schmitt from TERC (the Technical Education Resource Center) in
Cambridge, MA, and her colleagues, Veronica Hall and Andy Pate
Shirley Gilfether, Robin Warner, and Deborah Foucault from the Hampshire
Education Collaborative in Northampton, MA
Lynn Yanis (The Writer For You), who synthesized the collective wisdom and
resources, listened and incorporated many voices, and wrote, edited, and
designed the work to create a coherent and useful product
We especially want to recognize the Department ofYouth Services, its students,
and the teachers and program staff who work every day to bring clarity and
focus to the delivery of educational services in the DYS system.We offer special
thanks to the following teachers from the 2006 DYS Summer Academy, all of
whom provided comments and valuable feedback on an early draft of this work:
James Acheampong, Nat Alderman, Jennifer Avezzie, Melanie Blood, Carolyn
Davies, Guri Dura, Jenetha Gardiner, Diane Jardin, Gayle Kelly, Robert McKnight,
Dennis Mone, Gopesh Pandey, Ron Perrott,William Sheehan,Arthur Tunnessen,
John Vencelette, and Rui Wang—thank you.
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