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Learning Standards for

Mathematics, Science, and Technology

Revised Edition March 1996

THE UNIVERSITY OF THE STATE OF NEW YORK Regents of The University

Carl T. Hayden, Chancellor, A.B., J.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . Elmira Louise P. Matteoni, Vice Chancellor, B.A., M.A., Ph.D.. . . . . . . . . . . . . . Bayside Jorge L. Batista, B.A., J.D. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . Bronx J. Edward Meyer, B.A., LL.B.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chappaqua R. Carlos Carballada, Chancellor Emeritus, B.S. . . . . . . . . . . . . . . . . . . Rochester Norma Gluck, B.A., M.S.W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New York Adelaide L. Sanford, B.A., M.A., P.D.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hollis Walter Cooper, B.A., Ph.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rochester Diane O’Neill McGivern, B.S.N., M.A., Ph.D. . . . . . . . . . . . . . . . . . . . . . . Staten Island Saul B. Cohen, B.A., M.A., Ph. D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New Rochelle James C. Dawson, A.A., B.A., M.S., Ph.D. . . . . . . . . . . . . . . . . . . . . . . . . . Peru Robert M. Bennett, B.A., M.S.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tonawanda Robert M. Johnson, B.S., J.D.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lloyd Harbor Peter M. Pryor, B.A., LL.B., J.D., LL.D.. . . . . . . . . . . . . . . . . . . . . . . . . . . Albany Anthony S. Bottar, B.A., J.D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Syracuse Merryl H. Tisch, B.A., M.A.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .New York

President of The University and Commissioner of Education RICHARD P. MILLS

Executive Deputy Commissioner of Education THOMAS E. SHELDON

Deputy Commissioner for Elementary, Middle, Secondary, and Continuing Education

JAMES A. KADAMUS

Assistant Commissioner for Curriculum, Instruction, Assessment, and Innovation EDWARD T. LALOR

Coordinator of Curriculum and Instruction ROSEANNE DEFABIO

The State Education Department does not discriminate on the basis of age, color, religion, creed, disabil- ity, marital status, veteran status, national origin, race, gender or sexual orientation in its educational pro- grams, services and activities. Portions of this publication can be made available in a variety of formats, including braille, large print or audio tape, upon request. Inquiries concerning this policy of nondiscrimi- nation should be directed to the Department’s Office for Diversity, Ethics, and Access, Room 152, Education Building, Albany, NY 12234.

iii

CONTENTS

Acknowledgments vi Introduction v

LEARNING STANDARDS FOR MATHEMATICS, SCIENCE, AND TECHNOLOGY AT THREE LEVELS 1

SAMPLES OF STUDENT WORK RELATED TO THE STANDARDS 67

Acknowledgments

iv

In addition to the people recognized in the Preliminary Draft Framework for Mathematics, Science, and Technology, we are grateful to the many writers whose work is reflected in this edition. Listed below are the names of those writers. Patricia L. Barker, Newburgh Free Academy Betty McKnight, Consultant, Trumansburg, New York George E. DeBoer, Colgate University Marian Berke Sloane, New York City Schools John Weil, Fishers Island School Glen Botto, Arlington High School Nancy C. Bryan, North JHS, Newburgh Myron Rock, Sewenhaka High School Irene Jovell, Niskayuna High School Merri J. Earl, Chenango Forks Central Schools

Jeffrey J. Hering, Cortland JSHS Joan Dodway, Deposit JSHS Bruce Tulloch, Bethlehem CSD Dave Ferguson, SUNY Stony Brook Dorothy Rosenthal, Consultant, Indian Lake, New York William Peruzzi, NYSTEP Theodore Bredderman, SUNY Albany Tom Liao, SUNY Stony Brook James Tien, Rensselaer Polytechnic Institute Monique Havasy, Graduate Student, SUNY Albany

We are also grateful to the many teachers from throughout New York State who contributed performance tasks and samples of student work for possible inclusion in this revised edition. Listed below are the names of those educators whose materials appear in this document.

Helen Sultanik, Yeshiva of Flatbush Sandra LaTourelle, Ausable Valley CSD Glen Botto, Arlington High School Neal Swernofsky, Island Park UFSD Todd Kelly, Avon CSD David Propp, New Rochelle City SD Tom Venezio, North Colonie CSD

Shelley Shanok, Fort Hamilton High School, NYC James Luddington, Gananda Central School Rose Ambrosino, Smithtown Middle School Mary Ann Harasymowycz, Transit Middle School,

Williamsville Richard Beeler, Niskayuna High School

New York State Education Department Staff:

Major Capers Jan Christman Roseanne DeFabio Michael Hacker Virginia Hammer Barry Jameson Ed Lalor Benjamin Lindeman Toni Meyer Philip Pillsworth Lynn Richbart Elise Russo

Project Editor:

Susan Siegel

Introduction

v

This revised edition of the Learning Standards for Mathematics, Science, and Technology incorporates changes to the content standards and performance indicators based on extensive review by the public. As educational practice improves, these standards will continually be revised.

New in this edition are samples of student work, along with teachers’ comments on the work. The examples are intended to provide some ideas of tasks that support attainment of the performance standards. They are not mod- els of excellence. Rather, they vary in degree of achievement. Some are at the “acceptable” level; others are at a “more proficient” level. It is important to remember that these are just suggestions of ways that students can demonstrate progress toward achieving the standards.

The State Education Department will continue to collect and publish samples of student work. As teachers become more familiar with the standards and students become more proficient in meeting them, the level of the performance standards and content standards will continue to rise.

Taken together, the content standards and the performance standards define the learning standards for students in mathematics, science, and technology.

The Board of Regents recognizes the diversity of students in New York State, including students with disabili- ties, students with limited English proficiency, gifted students, and educationally disadvantaged students, and has made a strong commitment to integrating the education of all students into the total school program. The stan- dards in the framework apply to all students, regardless of their experiential background, capabilities, develop- mental and learning differences, interests, or ambitions. A classroom typically includes students with a wide range of abilities who may pursue multiple pathways to learn effectively, participate meaningfully, and work towards attaining the curricular standards. Students with diverse learning needs may need accommodations or adaptations of instructional strategies and materials to enhance their learning and/or adjust for their learning capabilities.

vi

Standard 1: Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions.

Standard 2: Students will access, generate, process, and transfer information using appropriate technologies.

Standard 3: Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.

Standard 4: Students will understand and apply scientific concepts, principles, and theories pertaining to the physical setting and living environment and recognize the historical development of ideas in science.

Standard 5: Students will apply technological knowledge and skills to design, construct, use, and evaluate prod- ucts and systems to satisfy human and environmental needs.

Standard 6: Students will understand the relationships and common themes that connect mathematics, science, and technology and apply the themes to these and other areas of learning.

Standard 7: Students will apply the knowledge and thinking skills of mathematics, science, and technology to address real-life problems and make informed decisions.

Learning Standards for Mathematics, Science, and Technology at Three Levels

1

1. Abstraction and symbolic representation are used to communicate mathematically.

Students: • use special mathematical notation and symbolism to

communicate in mathematics and to compare and describe quantities, express relationships, and relate mathematics to their immediate environments.

This is evident, for example, when students:

▲ describe their ages as an inequality such as 7 < ❒ < 10.

2. Deductive and inductive reasoning are used to reach mathematical conclusions.

Students: • use simple logical reasoning to develop conclusions,

recognizing that patterns and relationships present in the environment assist them in reaching these conclusions.

3. Critical thinking skills are used in the solution of mathematical problems.

Students: • explore and solve problems generated from school, home,

and community situations, using concrete objects or manipulative materials when possible.

1. Th