MATH Teaching in DYS Schools An Instructional Guide for Educators in the Massachusetts Department of Youth Services October 2006 • g{x VÉÅÅÉÇãxtÄà{ Éy `táátv{âáxààá 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 2 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 6 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 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 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 66 67 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. 68 69 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 70 71 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 72 73 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 74 75 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 78 79 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 80 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 84 85 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 86 87 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 88 89 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 90 91 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 r ea 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 mobil i ty SIMILARITY, SCALE, & PYTHAGOREAN THEOREM PULLING IT ALL TOGETHER 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 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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