DOCUMENT RESUME ED 412 104 SE 060 633 Crosswalks: PACESETTER Mathematics and the National TITLE Standards. College Board, New York, NY. INSTITUTION 1997-00-00 PUB DATE NOTE 24p. PUB TYPE Reference Materials General (130) MF01/PC01 Plus Postage. EDRS PRICE DESCRIPTORS Equations (Mathematics); *Functions (Mathematics); Logarithms; Mathematical Applications; *Mathematical Concepts; *Mathematical Models; *Mathematics Curriculum; Mathematics Education; Mathematics Skills; Matrices; Polynomials; Secondary Education; Trigonometry ABSTRACT The National Council of Teachers of Mathematics (NCTM) has published three documents that propose standards related to mathematics education called "The Curriculum and Evaluation Standards for School Mathematics" (1989), "The Professional Standards for Teaching Mathematics" (1991), and "The Assessment Standards for School Mathematics" (1995). The Pacesetter mathematics program and the course through which it is implemented, Precalculus through Modeling, embody the standards related to secondary school mathematics set forth in these documents. This booklet gives a detailed explanation of the relationship between these standards and Pacesetter Mathematics. Unit 1 is based on students' previous experience with linear equations and focuses on problems associated with real-life situations to introduce mathematical modeling. The seconaaLy focus is o the concept of function in general mathematics. Unit 2 examines multiplicative changes in art, finance, and social sciences. Sample activities include developing a savings plan for a person who has won a lottery and developing a savings plan for buying a car. In Unit 3 students build from quadratic and cubic equations to investigate the properties of power, polynomial, and rational functions. The sine, cosine, and tangent functions are introduced in Unit 4 which also utilizes graphing calculators. Unit 5 covers modeling with matrices, and Unit 6 addresses modeling with various other functions. (AIM) ******************************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. ******************************************************************************** CROSSWALKS: Pacesetter Mathematics and the National Standards PERMISSION TO REPRODUCE AND U.S. DEPARTMENT OF EDUCATION Office of Educational Research and Improvement DISSEMINATE THIS MATERIAL DUCATIONAL RESOURCES INFORMATION HAS BEE GRANTED BY CENTER (ERIC) This document has been reproduced as rued from the person or organization originating it. Minor changes have been made to improve reproduction quality. TO THE EDUCATIONAL Points of view or opinions RESOURCES stated in this document do not necessarily INFORMATION CENTER (ERIC) represent official OERI position or policy. BEST COPY AVAILABLE Educational Excellence for All Students 7)EOL9Co; 7)-5 Crosswalks: PACESETTER Mathematics and the National Standards Crosswalks: PACESETTER Mathematics and the National Standards College Entrance Examination Board, New York 5 Pounded i1. 1099, 1-.11,- College Board is a national, nonprofit membership association of schools, colleges, and other eciticati,,i-al org? nations working together to help students succeed in the transition from school to college. The Board meets the diverse needs of schools, colleges, educators, students and parents through the development of standards of excellence; by provid- ing programs and services in guidance, assessment, admission, placement, financial aid, and teaching and learning; and by conducting forums, research, and public policy activities. In all of its work, the Board promotes universal access to high standards of learning, equity of opportunity, and suf- ficient financial support so that every student has the opportunity to succeed in college and work. Copyright ©1997 by College Entrance Examination Board. All rights reserved. College Board, Pacesetter, and the acorn logo are registered trade- marks of the College Entrance Examination Board. Printed in the United States of America. Crosswalks PACESE1TER® Mathematics and the National Standards "Pacesetter Mathematics is a working example of a standards-based mathematics curriculum. Its focus on the featured processes of problem solving, reasoning, communicating, and connecting are immediately evident on investigation of the materials designed for student use" (Introduction, Precalculus through Modeling, Teacher's Ed.). Introduction The National Council of Teachers of Mathematics has published three docu- ments in the past few years that propose standards related to mathematics educationthe Curriculum and Evaluation Standards for School Mathematics (1989), the Professional Standards for Teaching Mathematics (1991), and the Assessment Standards for School Mathematics (1995). The Pacesetter mathemat- ics program and the course through which it is implemented, Precalculus through Modeling, embody the standards related to secondary school math- ematics set forth in these documents. This document will give a detailed explanation of the relationship between these standards and Pacesetter Mathematics. PACESETTER Mathematics Units and Task Sets UNIT 1Modeling with Linear Functions This unit is based on students' previous experience with linear equations and focuses on problems associated with real-life situations as a way of intro- ducing students to mathematical modeling. A second area of focus is the concept of function as a general mathematical idea apart from any particu- lar application. Stacks of CupsStudents are asked to generate a mathematical model for the height of a box that is to contain a stack of a certain number of cups. They must generate their own data through experimentation. The topics addressed in this task set are: building a linear model from student-generated data, graphing linear functions, slope, y-intercept, equations of lines, points of intersection, discrete value functions, direct variation, and arithmetic sequences. Crosswalks: PACESETTER Math and the National Standards 1 The Mile RunStudents are asked to generate a model to help pre- dict when the record time for the mile run will reach a certain value given a set of data for previous records. The topics addressed in this task set are: using "dirty data," lines of best fit, making predictions from data, slope, intercepts, equations of lines, concept of linear regression, and median-median line (optional). Ships in the FogStudents are asked to determine whether two ships that are being tracked on a screen will collide. In the process, they will generate a mathematical model for the paths of the ships and use their model to answer questions about the paths of the ships. The topics addressed in this task set are: linear relationships defined parametri- cally and graphs of these relationships, motion in the plane, intersec- tion of lines, distance formula, and inverse functions. Temperature ScalesStudents are asked to examine the relationships among the Celsius, Fahrenheit, and Kelvin temperature scales. The topics addressed in this task set are: slope, intercepts, inverse func- tions, composition of functions, and translations of graphs. UNIT 2Modeling with Exponential and Logarithmic Functions This unit requires students to examine multiplicative changes in the areas of art, finance, and social sciences. As contrasted with the arithmetic growth seen in Unit 1, the regularity of geometric growth is a constant takes place from one setting or ratio between succ.c.Gsivc rhan o-e ttrr_15 time frame to the next. The Square Craft DesignStudents are asked to develop a model that can be used to determine the sizes of the triangular pieces in a craft design. The topics included are: development of an exponential func- tion through student-generated data, isosceles right triangles, ratio, similarity, scale, and area. The Pennsylvania LotteryStudents are asked to develop a savings plan for a person who has won a lottery. The topics included are: compound interest (with varying compounding periods), exponen- tial growth, the ratio property of exponential functions, graphs of exponential functions, and nominal and effective interest rates. A Powerful FunctionThis task set is used to convey to the student the importance of logarithmic functions as inverses of exponential functions in solving exponential equations. Students use the calcula- Crosswalks: PACESETTER Math and the National Standards 2 a for as a trial-and-error tool to first make a connection between expo- nents and logarithms. Discovery exercises lead to the properties of logarithms. The topics included are: common logarithms, other base logarithms and the change-of-base formula, logarithmic functions, inverse functions, composition of functions, and solution of exponen- tial equations. Population GrowthStudents are asked to estimate when the world's population will reach ten billion. Topics included are: making predic- tions using "dirty data," exponential functions, and the concept of exponential regression. Saving to Buy a CarStudents are asked to develop a savings plan that would provide enough money to buy a car. The topics included are: parameters of an exponential function, sum of a geometric series, graphs of exponential functions, and analysis of these graphs. UNIT 3Modeling with Polynomial and Rational Functions This unit extends the ideas of modeling beyond linear and exponential functions. Building from quadratic and cubic equations, students investigate the properties of power, polynomial, and rational functions. FencesStudents are asked to determine for a customer the largest possible rectangular area for a dog enclosure, given a certain length of fencing. The topics included are: quadratic functions and their graphs; the relationship among the graphs of these functions, their roots, and the discriminant; the relationship between area and perimeter; intro- duction to rational functions; and maximizing volume. Highway SafetyStudents are asked to determine how many feet it would take to bring a car that is traveling at a speed of fifty miles per hour to an emergency stop, given an average reaction time. The topics included are: modeling with linear and quadratic functions, data analysis, optimization, dimensional analysis, and analysis of the param- eters of a function. Planning a Summer CampStudents are asked to determine the per camper cost for children to attend a summer camp, given certain conditions. The topics included are: rational functions, translations of rational functions, vertical and horizontal asymptotes, and inverse functions. 3 Crosswalks: PACESETTER Math and the National Standards Graphs of Rational FunctionsStudents are asked to examine and interpret the graphs of various forms of rational functions. The topics included are: increasing and decreasing functions, domain, bounded- ness, end behavior, asymptotes, graphs of sums and quotients of func- tions, and infinite geometric series. The Ideal Traffic PatternStudents are asked by the City Council to design a highway traffic flow pattern that meets the requirements of several special interest groups. This task set is based on the informa- tion found in the Highway Safety task set. The topics included are: optimization, dimensional analysis, graphical analysis of functions, and exploration of a rational function and a piecewise-defined function. Modeling with Trigonometric Functions UNIT 4 The sine, cosine, and tangent functions are introduced through periodic phenomena such as the horizontal and vertical projection of a rotating bicycle wheel. Students use graphing calculators to explore the graphs of these functions. Bicycle WheelsA two-wheeler with training wheels runs over a freshly painted parking strip. Students are asked to determine when the paint strip on the front wheel is the same height above the pave- ment as the paint strip on the training wheel. The graph of a sine function is developed. The topics included are: circular motion, right frecpiency. ver- triangle trigonometry, tical translation, phase shift, degree measure versus radian measure, and the graph of a cycloid, using parametric equations. Daylight HoursStudents are asked to determine how many daylight hours a specific city would experience on a specific date of the year. The topics included are: periodic phenomena, the sine function, parameter analysis, data analysis, geometry of the sphere, right trian- gle trigonometry, and connections to astronomy and geography. The Chocolate FactoryGiven certain specifications, students are asked to design a vat that is used as part of the mechanical stirring equipment in a chocolate factory. The topics included are: the unit circle, definition of sine and cosine, the Pythagorean theorem, similar triangles, right triangle trigonometry, law of cosines, angular velocity, and the intuitive meaning of boundedness (limits). The Discus ThrowStudents are asked to derive a formula that will and the National Standards 4 Crosswalks: PACESETTER Math 10