bluuebunny/crossref_metadata_2025 huggingface dataset:
Investigating Absolute-Value Equations with the Graphing Calculator
Authors: Horak Virginia M.
Published: November 2020

The National Council ofTeachers of Mathematics's Curriculum and Evaluation Standards for School Mathematics (1989) urges the implementation of technology in all areas of mathematics learning. In a high school algebra class, the graphing calculator can be a powerful aid in developing mathematical understandings. Equations involving absolute value are rich ground for graphical exploration. Students can apply their knowledge of graphs of lines and solution sets of simultaneous linear equations to enhance their conceptual understandings of these topics.
Implementing the “Curriculum and Evaluation Standards”: Recognizing and Recording Reform in Mathematics: New Questions, Many Answers
Authors: Ferrini-Mundy Joan , Johnson Loren , Smart James R.
Published: November 2020

NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and its Professional Standards for Teaching Mathematics (1991) lend possible direction and meaning to the reform effort in mathematics education that is sweeping the country. The documents have been widely disseminated and discussed, and anecdotal evidence indicates that teachers of mathematics are seeking ways to enact the ideas contained in the standards documents. These documents are also inspiring the development of standards in other disciplines. But a number of questions are being raised as schools, districts, states, and provinces attempt to incorporate these Standards in changing their curriculum and pedagogy.
Using Similarity to Find Length and Area
Authors: Sandefur James T.
Published: November 2020

Generally in high school, students learn how to find the length of straight lines and can use that information to find the perimeter and the length of the diagonal of simple polygons. They also learn the formula relating the diameter and the perimeter of a circle, and they know how to find the areas of circles, rectangles, triangles, and, possibly, regular n-gons. But the curves whose length and the figures whose area can be calculated by students are relatively mundane, and the methods are straightforward. The techniques used do not lend themselves easily to combining algebra and geometry
Sharing Teaching Ideas: The Real-Number Box
Authors: Severson Jean
Published: November 2020

The first lesson I taught in second-year algebra involved classifying real numbers. I wanted to take students from the concrete, to the pictorial, to the abstract. The pictorial and the abstract parts of the lesson seemed obvious. Diagram real numbers. Discuss and use classroom exercises. I wrestled with how to make the concepts concrete. My answer was the real-number box.
Surprising Results Using Calculators for Derivatives
Authors: Goetz Albert , Kahan Jeremy
Published: November 2020

“When is the numerical derivative obtained on the calculator greater than the actual derivative and when is it smaller?” A group of college professors and high school mathematics teachers attending a workshop seemed not to know the answer. The question was posed by Deborah Hughes-Hallett, then of Harvard University, who is currently at the University of Arizona, at the first Technology Intensive Calculus for Advanced Placement (TICAP) institute at Clemson University in spring 1992.
Connecting Geometry and Algebra: Geometric Interpretations of Distance
Authors: Crites Terry W.
Published: November 2020

Activities that demonstrate connections between geometry and algebra should be an Important part of the secondary school curriculum (NCTM 1989). Teachers should design lessons that allow students to investigate, discuss, and strengthen the natural ties that exist between these two areas of mathematics. Exploring topics from both a geometric and an algebraic perspective should be a constant feature of any mathematics curriculum
“What If…?” A Case in Point
Authors: Friel James O. , Gannon Gerald E.
Published: November 2020

The following problem appears as a “brainteaser” in the fifth edition of Mathematics for Elementary School Teachers by Billstein, Libeskind, and Lott (1993, 298).1t is an excellent problem for students, since it can be solved nicely either by using a straightforward algebraic approach or by working backward.
Sharing Teaching Ideas: Mathematics Notebooks
Authors: Conrath Patrice L.
Published: November 2020

Students are often surprised to find that mathematics course work is a daily task. To help students organize the seeming flood of homework, projects, and notes, I have found that requiring them to keep notebooks has been very useful.
Activities: A Visual Approach to Deductive Reasoning
Authors: Dyke Frances Van
Published: November 2020

One of the primary goals of education is to enhance students' ability to reason deductively. The capability to think logically is needed in every discipline, and it is particularly important in mathematics. Standard 3 of the NCTM's curriculum standards for grades 9–12 recommends that the “curricu lum should include numerous and varied expetiences that reinforce and extend logical reasoning skills” (NCTM 1989, 143).
The Square-Patio Problem
Authors: Lowther Martha H.
Published: November 2020

The problem discussed in this article was presented at the Delaware Teacher Enhancement Partnership Workshop in July 1994. The problem incorporates manipulatives; pattern building; graphing calculators, such as the TI-82; and matrix applications and functions and can be adapted to any level from middle school through high school. It covers the topics of geometry, algebra, discrete mathematics, and technology and can be explored with cooperative learning. The matrix application is emphasized in this presentation, but other features of the problem, including polynomials and the use of lists on the TI-82, are also worthy of attention. A worksheet containing the problem is included for students.
Trap a Surprise in an Isosceles Trapezoid
Authors: Housinger Margaret M.
Published: November 2020

Geometry is a most exciting subject, filled with all sorts of properties and descriptions of the world around us and the worlds we invent in our minds. Areas and volumes constitute a major part of any geometry course. These topics are always enjoyable, since they can be modeled against actual objects. Unexpectedly, I had the opportunity to feel like a real geometer when I began some routine work.
Illustrating Mathematical Connections: A Geometric Proof of Euler's Theorem
Authors: Frye Erin K. , Glidden Peter L.
Published: November 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) calls for teachers to emphasize mathematical connections, promote mathematical reasoning, and help students become better problem solvers. If teachers are to achieve these goals, they need compelling examples, problems, and theorems that address all these elements.
The Pentagon Problem: Geometric Reasoning with Technology
Authors: Zbiek Rose Mary
Published: November 2020

We value the use of technology in mathematics learning and teaching, and we want students to reason and to explore mathematical ideas in their mathematics courses. In recent years, such computing tools as The Geometer's Sketchpad (1991) and Cabri Geometry II (1994) allow us to devise and operate on geometric figures similarly as symbolic manipulators allow us to work with algebraic expressions. In this article, we call these tools figure manipulators. These tools make it possible for students to explore and connect geometric ideas from synthetic, analytic, and transformational perspectives. Yet we wonder how we can actually get this synthesis to happen in our classrooms. Our doubting colleagues, and we, question the effects of such experiences on our students' understanding of mathematics in general and of geometry in particular.
Activities: An Interest in Radioactivity
Authors: Kennedy Jane B.
Published: November 2020

Introduction: This activity explores exponential growth and decay, emphasizing the paired concepts of doubling and half-life. Exponential growth is derived from actual computations to obtain compound interest, whereas exponential decay is modeled by the use of “radioactive” dice. The activity is based on the concept of the differentiated core curriculum, which asserts that all students should be guaranteed equal access to the same curricular topics but recognizes that all students may not explore the content to the same depth or at the same level of formalism.
Multiple Connections
Authors: Zbiek Rose Mary
Published: November 2020

What makes a good scenario for a fruitful experience with connections? The following example illustrates a web of connections that students might make in a real-world scenario that begins with familiar and fundamental concepts: perimeter and area of rectangles.
New York City
Authors: De Graff Forrest A.
Published: November 2020

To secure the information asked for in questions (b) and (c) of today’s program, a questionnaire embodying them was addressed to fifteen high schools and five colleges in New York City, as well as to several representative institutions in nearby municipalities whose industrial problem seemed to invite adjustments of mathematical courses. Although several replies are lacking, an effort has been made through personal conferences and perusal of catalogs to supply the missing information.
Entrance Requirements Again
Authors: Keller S. S.
Published: November 2020

The writer has little hope of seeing and not the least desire to see this protean pedagogical question of entrance requirements completely answered to the purring satisfaction of everybody or of even a majority.
Reader Reflections
Authors: Lewis Geoffrey J. , Metro Kathy
Published: November 2020

The “Soundoff!” in the November 1996 issue of the Mathematics Teacher, pages 426–27, “Who Is Dick Askey and Why Is He So Upset about the Sandards?”
When Knottiness Is Fitting Behavior
Authors: Fosnaugh Linda S. , Harrell Marvin E. , Fosnaugh Timothy
Published: November 2020

Before the invention and affordability of modern depth finders, recreational anglers often tied knots in the boat's anchor rope to determine the depth of the river or lake. One reason for this practice is that different types of fish tend to prefer different depths of water. The knot-tying practice is still common among many old-time fishermen. We have found that knot tying provides a imple hands-on context for student modeling in algebra.
(Naturally) Numbers Are Fun
Authors: Crossfield Don
Published: November 2020

About six years ago, I put in my classroom large colored banners of the sequences shown here. This article lists some ways that these sequences have gradually been incorporated into my lessons, as well as my observations about the caliber of thinking and questioning that my students are developing as they learn appropriate ways to use these numbers.
Reader Reflections
Authors: Lynch Dennis
Published: November 2020

Is block scheduling good for mathematics classes? Yes! At John Stark Regional High School. in Weare, New Hampshire. we have been educating our students under the block-scheduling format for more than five years. The teachers love it and the students love it.
Activities: Patterns and Puzzles for Pyramids and Prisms
Authors: DeTemple Duane , Miedema Allen
Published: November 2020

Teacher's Guide: The construction projects, puzzles, and experiments presented here provide hands-on experiences in spatial visualization and problem solving for an important class of three-dimensional figures: pyramids and prisms. Students will enjoy creating physical models and performing experiments with the models, but the main goals of the activities are to develop students' geometric intuition and build a concrete foundation on which abstract principles can be grounded.
Solutions to Calendar
Authors: McLoughlin John Grant
Published: November 2020

Problems 1, 6, 26, and 27 were contributed by Harry Simon, 701 Viola Street, Eunice, LA 70535-4339. Problems 2, 7–11, 18, 19, 22, 25 and 31 were prepared by Bob Tex Kenney, P. 0. Box 454, Saipan, MP 96950. Problems 3, 4, 5, 21, and 23 were adapted from Mathematical Quickies by Charles W. Trigg (Mineola, NY: Dover Publications, 1985). Problems 12–14 and 20 were offered by Morris Jack DeLeon, Florida Atlantic University, Boca Raton, FL 33431. Problem 15 was submitted by Enrico Uva, Outreach Schools, 1741 de Biencourt, Montreal, PQ H4E 1T4. Problems 16, 17, and 24 were adapted from The Moscow Puzzles: 359 Mathematical Recreations by Bons A. Kordemsky (Mineola, NY: Dover Publications, 1992). Problems 29 and 30 were contributed by Mark Harbison, 5737 College Avenue, #23, San Diego, CA 92120.
Technology Reviews and Publications
Authors: Burke William , Kick Richard
Published: November 2020

Prices on software, books. and materials are subject to change. Consult the suppliers for the current prices. The comments reflect the reviewer's opinions and do not imply endorsement by the National Council of Teachers of Mathematics.
Making Music with Mathematics
Authors: Fernandez Maria L.
Published: November 2020

Have you ever wondered how to bring music into your mathematics lessons? Music can be a rich context for facilitating mathematical connections.
The Big Loser
Authors: Marks Daniel
Published: November 2020

The identity of the team in greatest jeopardy of becoming the big loser is the subject of this article. This article explores several facts about the big loser, offering them in a hierarchy that may be appropriate for creating various short– and long–term projects for a high school mathematics class.
Mathematics as Art: The Missing Standard
Authors: Rogers James R.
Published: November 2020

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world.
Using a Digital Camera to Verify Quadratic Behavior
Authors: Shultz Harris S.
Published: November 2020

This activity presents practical difficulties, not the least of which is the proximity of a large amount of spraying water to the students marking the points on the wall. This difficulty can be circumvented by photographing the stream of water. In particular, the ability of many digital cameras to communicate electronically with personal computers simplifies the task of recording several coordinates of the stream of water.
The Oak Leaf: Connecting Geometry and Biology
Authors: Snyder Judy
Published: November 2020

An activity that involves using geometry, statistics, and computers to interpret data about the leaves of a tree. Specifically, students determined the comparative area, weight, and photosynthetic activity of leaves for leaves growing on the outer part of an oak tree, that is, sun leaves, and those found on the inner part of the tree, or shade leaves. The students were involved in actual research, and mathematics was an integral part of their research.
Sharing Teaching Ideas: When Will I Ever Use This Stuff, Anyway?
Authors: Austin Suzanne S. , Berceli Charlotte L. , Mathews Sarae
Published: November 2020

As instructors of an intermediate-algebra course at a large urban community college, we decided to use a service-learning project to help students understand and appreciate the mathematics that they were studying. This outcome transcended the classroom and best motivated mathematics students by helping them realize how important the subject is to their futures (Jenrette 1994).
Statistics in Context
Authors: Watson Jane M.
Published: November 2020

Judging statistical claims in social contexts is fundamental to statistical literacy. This article uses a particularly contentious newspaper report that makes a cause-and-effect claim as the basis for discussing this important aspect of statistical understanding. The issue's relevance across the school curriculum is shown by extracts from curriculum documents. Teachers need to structure experiences to build ability to question claims made without proper justification. This article suggests a hierarchy to help teachers plan for and assess student learning in this area, and it closes with a plea for teachers to cooperate across subjects to achieve results.
Algebra for All: Reunion of Broken Parts: Experiencing Diversity in Algebra
Authors: Lesser Lawrence M.
Published: November 2020

At this time in American education, many educators (e.g., Manning [1995]) continue to struggle with the balance of highlighting differences and highlighting common ground among individuals with diverse backgrounds. Giving full educational opportunities to all students, however, not only is the right thing to do as a matter of justice but also enriches the educational experience of all individuals involved. The National Council of Teachers of Mathematics (1997, 5) explains one reason that these opportunities are essential for the algebra course: “In recent years, [algebra] has become a ‘gateway’ course to higher education, particularly for minority students. Those students who steer away from algebra early often forfeit some of their options for the future.”
Media Clips: February 2000
Authors: Westegaard Susanne K.
Published: November 2020

We welcome and encourage submissions to this department, which will comprise resources that teachers can use immediately or file away for future use. The assignments generated by these entries may range from starters for an entire class to fullblown projects for optional individual pursuit. Each lesson, though, should help convince our students that mathematics is useful, current, and newsworthy. Please send submissions to “Media Clips,” Mathematics Teacher, 1906 Association Drive, Reston, VA 20191-9988.
Sharing Teaching Ideas: Journal Writing in the Mathematics Classroom: A Beginner's Approach
Authors: Williams Nancy , Wynne Brian
Published: November 2020

Today's mathematics educators are advocating methods of assessment other than frequent tests, quizzes, and daily worksheets. In fact, the NCTM's Curriculum and Evaluation Standards states,“The assessment of students' ability to communicate mathematics should provide evidence that they can express mathematical ideas by speaking, writing, demonstrating and depicting them visually” (1989, 214). One alternative form of assessment that incorporates these standards is journal writing
Using The Geometer's Sketchpad to Visualize Maximum-Volume Problems
Authors: Purdy David C.
Published: November 2020

An underlying tenet of the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and other movements toward reform in school mathematics is breaking down content barriers between traditional mathematical topics, with the goal of teaching mathematics as a logically interconnected body of thought. As teachers move toward integrating the various areas of mathematics into traditional courses, problems that were once reserved for higher courses, for example, precalculus and calculus, now surface earlier as interesting explorations that can be tackled with such tools as the graphing calculator. One such problem is the well-known maximum-volume-box problem. Although this problem and related optimization questions have been common in advanced algebra, precalculus, and calculus textbooks, they have only recently found their way into high school geometry textbooks, including Discovering Geometry: An Inductive Approach (Serra 1997).
Sum of Songs: Making Mathematics Less Monotone!
Authors: Lesser Lawrence Mark
Published: November 2020

Songs in the mathematics classroom can be fun and functional; they can supply motivation and mnemonics. Students and teachers with even minimal musicianship can enrich their connections to mathematics by using existing songs, writing raps, or writing new words for existing tunes.
The Spaghetti Problem Problem
Authors: Ippolito Dennis
Published: November 2020

Who would think that all this mathematics could come from a strand of spaghetti? In attempting to solve this problem we used such tools as algebra, geometry, probability, statistics, computer simulations, and calculus.
A Triangle Divided: Investigating Equal Areas
Authors: Scher Daniel
Published: November 2020

Sometimes beneath a simple problem statement lies unexpected mathematical depth. Consider this challenge from the Connected Geometry curriculum (Education Development Center 2000): “Find as many ways as you can to divide an arbitrary triangle ABC into four equal-area triangles.” Before reading further, try solving the problem yourself.
Mathematics in Search of History
Authors: Barry Donald T.
Published: November 2020

The history of mathematics is used in a variety of ways to liven up a classroom and give meaning to a lesson. We retell anecdotes, require that students research the lives of mathematicians, encourage students to study the attempts to solve problems of historical interest, and suggest that students discover links between the historical development of science and the development of mathematics. I worry, however, that we are primarily treating the history of mathematics as a fixed entity that consists solely of a set of facts, not as a fluid field within which lively debate occurs and emphases shift over time.
Kepler and Wiles: Models of Perseverance
Authors: Shotsberger Paul G.
Published: November 2020

If you have seen the videotape The Proof or read the companion book by Simon Singh (1997) called Fermat's Enigma, you know that the story of Andrew Wiles's journey toward the proof of Fermat's last theorem is a remarkable tale of hope, disappointment, persistence, and ultimate triumph. Discovered by Pierre Fermat around 1637, the theorem is simple to state: “The conjecture that xn + yn = zn, where n > 2, has no solution with x, y, and z positive integers” (James and James 1992, p. 163). Yet the proof of this well-known theorem eluded generations of mathematicians. Wiles spent nearly a decade attempting to prove the theorem before he finally succeeded in 1994. The videotape and the book each offer an all-too-rare glimpse into the private struggles that preceded the more formal presentation of a finished product.
The Sumario Compendioso: The New World's First Mathematics Book
Authors: Gray Shirley B. , Sandifer C. Edward
Published: November 2020

When were mathematics books first written in North America? Some students who are asked that question will suggest that the settlers in Jamestown began writing mathematics books after they arrived in 1607. Others will think that North Americans started writing mathematics books right after signing the Declaration of Independence in 1776. A rare student indeed remembers that the Spanish American colonies were established much earlier than the English or French colonies. Few will guess anything close to the correct answer: the New World's first mathematics book, the Sumario Compendioso, was published in 1556 in Mexico City by Brother Juan Diez.
Thinking and Writing Mathematically: Achilles and the Tortoise as an Algebraic Word Problem
Authors: Martinez Joseph G. R.
Published: November 2020

Discusses author's view of word problems in an Algebra class and describes an imaginative context with a thinking and writing exercise he developed for classroom use.
Stretching Pythagoras around the Corner: Linking and Modeling in Precalculus
Authors: Collingwood David H. , Stor Marilyn
Published: November 2020

Allyson and Adrienne have connected their ankles with a bungee cord. They start at the same location near the outside corner of a building and move in the directions indicated in figure 1, which depicts the situation from above
Dividable Triangles—What Are They?
Authors: Leikin Roza
Published: November 2020

Introduces a triangle for which many different problems can be posed on the basis of one unusual property. Three problems on this property are posed, along with solution strategies that can be used in classroom activities.
Visualizing the Complex Roots of Quadratic and Cubic Equations
Authors: Lipp Alan
Published: November 2020

A method for visualizing the complex roots of a polynomial equation, thereby making them a little more “real” and a little less “imaginary.”
Activities for Students: Searching Families as a Source of Surprise
Authors: Winicki-Landman Greisy
Published: November 2020

“Mathematics at all levels is a box full of surprises,” according to Movshovitz-Hadar (1988, p. 34). In general, however, students are not aware of this aspect of the subject.
Sharing Teaching Ideas: Mathematical Modeling: Compound Functions and the IRS Tax Rate Schedules
Authors: Wolbert William J.
Published: November 2020

When a teacher finds a real-life application of mathematics, especially one to which every student in a class can relate, expounding on that application is certainly worthwhile. This article focuses on the concept of compound functions, as presented in a second-year algebra context, and on an application of this concept using the tax rate schedules from 1999. Teachers can adapt this activity to reflect current tax information.
Privacy for the Twenty-First Century: Cryptography
Authors: Petras Richard T.
Published: November 2020

The Internet is everywhere today. E-mail is as common as a telephone call, and every company includes its Web site in its advertisements. The Internet economy was expected to surpass $1.3 trillion by summer 2000 (Must Read 1999a), and online retailing was expected to generate $36 billion in revenue in 1999, a 250 percent increase over 1998 (Must Read 1999b). In the present millennium, daily life at all levels will increasingly depend on the transmission over public lines of sensitive, private data, whether it is payment information or personal e-mail. Since “nearly 60% of US consumers think that transactions made via the Internet are unsafe” (Must Read 1999a), the ultimate success of the Internet economy depends on the ability to guarantee privacy on public systems. Cryptography and secure cryptosystems will help protect the privacy of all computer users, ranging from government and military organizations to the individual user at home.
Alternative Geometric Constructions: Promoting Mathematical Reasoning
Authors: Pandiscio Eric A.
Published: November 2020

Construction tools in most high school Euclidean geometry classes have typically been limited to a compass for drawing circular arcs and a straightedge for drawing line segments. The strengths of these tools include both mathematical precision and a long history of use. However, alternatives can provide fresh possibilities for engaging students in the mathematical reasoning that lies at the heart of traditional geometry (Gibb 1982; Robertson 1986). This article proposes that a single task completed with a variety of construction tools fosters a greater sense of mathematical contemplation than multiple tasks done with the same tool. The premises are simple: each tool fosters different mathematical ideas, and using multiple tools not only requires understanding of a greater breadth and depth of geometric concepts but also highlights the connections that exist among different ideas.
Project VISM--Visualization in Science and Mathematics
Authors: Kolvoord Bob , Foletta Gina
Published: November 2020

Visualizing data is a great gateway to exciting mathematics and science content for many students and teachers. However, all too often, training and professional development focus on all the details of a single tool, forcing the learners to wonder what happened to the forest while they were learning all about the tree. In Project VISM—Visualization in Science and Mathematics, we are pioneering new methods of staff development and helping educators see the connections among many of the exciting tools that are available.
Benjamin Banneker's Mathematical Puzzles
Authors: Mahoney John
Published: November 2020

enjamin Banneker was a scientist, scholar, astronomer, and mathematical wizard. An African American, he was born a free man in Maryland in 1731. Banneker was known and respected throughout America and Europe for his mathematics skills. Using only a pocket watch as a model, he built the first wooden striking clock in America. It kept accurate time for fifty-three years. President Jefferson had such high regard for Banneker that he chose him to assist in developing plans for laying out the streets of Washington, D.C. From 1792 to 1797, Banneker published a Farmer's Almanac, which contained his calculations of the positions of the sun, moon, and planets. These calculations enabled him to predict eclipses and weather conditions. Banneker died in 1806 on his Maryland farm.
Sharing Teaching Ideas: Getting to Know a Calculator's Numerical Limitations
Authors: Krumpe Norman , Keiser Jane
Published: November 2020

In a class of twenty or so students, finding three or four different kinds of graphing calculators in use at the same time is not at all unusual. Such variety poses challenges to everyone in the class. Differences in the way that various calculators handle very large and very small numbers—as well as “medium–sized” numbers with many significant digits—can lead to discrepancies when students compare their results with those obtained by other students. Teaching students to become aware of the causes of such discrepancies can help them take a more active role in interpreting their computational results and in working around the limitations of their calculators.
Projects: SimCalc Project
Authors: Kaput Jim
Published: November 2020

The SimCalc Project, partially supported for almost ten years by the National Science Foundation, is a research project involving colleagues from several universities, many middle schools, and high schools. Its goal is to use innovative but affordable technology and instructional materials to democratize access to the mathematics of change and variation. It includes, for example, ideas underlying calculus, beginning in the middle school through algebra and onward into university calculus. SimCalc approaches involve a growing mix of visually editable graphs that control dynamic simulations, visualization and data-import tools, and curriculum materials that use these software capabilities, as well as such traditional ones as function graphers. These strategies address the basic ideas of rate, slope as rate, linear functions, simultaneous conditions, interpreting graphs and modeling word problems (including more realistic ones than those that appear in most textbooks), the idea of average, mixture problems, equations, signed numbers and areas, variables and variation, and so on. They also include such ideas as periodicity, velocity and position, and rate and totals connections, including the slope and area ideas underlying the fundamental theorem of calculus.
On the Distribution of Primes
Authors: Johnson John M.
Published: November 2020

he distribution of primes throughout the natural numbers is a wonderful mystery that has always entertained mathematicians—professional and amateur, genius and ordinary—yet complete understanding has eluded their attempts. The names of those who have considered the problems discussed in this article and related problems form a “mathematics hall of fame” and span at least twenty-three centuries. This list includes Euclid, Euler, Fermat, Gauss, Legendre, and so on. This article highlights some of the well-known properties of primes and explores some properties that are not as wellknown but that are accessible to anyone who understands what a prime number is and who has a healthy amount of curiosity.
Calendar: April 2003
Authors: Benson Steve
Published: November 2020

Problem 1 was adapted from a problem in One Equals Zero and Other Mathematical Surprises, by Nitsa Movshovitz-Hadar and John Webb (Berkeley, Calif.: Key Curriculum Press, 1998). Problem 2 is adapted from a problem from the Mathcounts 1998–99 Handbook. Problems 3–14, 27, and 28 come from the mathschallenge.net Web site. Problem 15 is from the Math Forum's Problem of the Week Archives, available at mathforum.org/pow. Problems 16–25 came from—or were adapted from—problems at the Mathematics Problems and Warm-ups Web site, at www.geom.umn.edu/~lori/mathed /problems. Sources of the problems, as cited on the Web site, are as follows: problems 16, 18–21, and 25 are from Mathematical Games, by Marie Berrondo (Englewood Cliffs, N.J.: Prentice Hall, 1983); problems 17, 22, and 24 are from Fun with Brain Puzzlers, by L. H. Langley-Cook (Greenwich, Conn.: Fawcett, 1965); and problem 23 is from the Minnesota Math League. Problem 29 comes from the “Nets of Cubes” problem set from Problems with a Point, a National Science Foundation–supported project at Education Development Center, available at www2.edc.org/mathproblems.
Teaching with Technology: Dynamic Visualization and Proof: A New Approach to a Classic Problem
Authors: Scher Daniel
Published: November 2020

IN 2001–2002, THREE TEACHERS from Paul Robeson High School in Brooklyn, New York, joined an international collaboration to pilot U.S.-Russian Interactive Geometry (Armontrout et al. 2002) labs in their classrooms.
Sharing Teaching Ideas: The Triangle Technique for Solving Direct Variation Problems
Authors: Lee Raymond E.
Published: November 2020

Students encounter and have difficulty with problems that involve direct variation and joint variation. Many of these problems are called work problems, rate problems, or percent problems. However, many real-life situations involve quantities that change (vary) as a result of another quantity or quantities.
Media Clips: October 2003
Authors: Patrick Vennebush G.
Published: November 2020

“Media Clips” aims to offer readers contemporary, authentic applications of quantitative reasoning that are based on print or electronic media. We encourage submissions, which may include questions, as well as clips. We especially welcome examples of flawed quantitative reasoning. Clips may be in a text or graphic format, and clip sources may be either print or electronic media. Submissions must include a complete citation of the source, along with the full text of the cited material. If possible, submissions should also include a copy of the actual clip. Please send submissions to “Media Clips,” Mathematics Teacher, 1906 Association Drive, Reston, VA 20191-1502.
Projects: Students Building Models—Building Model Students
Authors: Foletta Gina , Becker Joyce
Published: November 2020

What shape do two galaxies take when they collide? How long does a newly introduced virus take to spread through a city the size of Chicago? Such questions are often difficult to answer because the phenomena are difficult to observe directly. The perceptible motion of galaxies occurs over time scales that are greater than the entire history of mankind; and it is thus impossible to truly observe even with a lifetime of study. Although the spread of a virus might be indirectly observable through reports at local hospitals, pinpointing an original carrier and tracking infected individuals is neither plausible nor particularly desirable.
Projects: Math Tools: The Software You Want with the Help You Need
Authors: Foletta Gina
Published: November 2020

The Math Forum has a new project to support the use of software for mathematics education. It is developing a community library where teachers, teacher educators, educational researchers, software developers, and students can all work and learn together about effectively using technology in the classroom.
Activities for Students: A-B-C, 1-2-3
Authors: Richardson Mary , Gabrosek John
Published: November 2020

This article describes an activity that revolves around a data set that students help create. The students use data about occurrences of letters in English text to study the relationship between the relative frequency of letters and the percent of Scrabble game tiles for the letter and the relationship between the relative frequency of a letter and the letter's Scrabble game-tile point value.
Building Your Own Regression Model
Authors: Horton Robert M. , Phillips Vicki , Kenelly John
Published: November 2020

Regression has rightfully assumed an important role in the secondary mathematics curriculum. Graphing calculators can perform many types of regression quickly and easily, and students can learn a great deal in analyzing the results.
Copernican Mathematics: Calculating Periods and Distances of the Planets
Authors: Rosenkrantz Kurt J.
Published: November 2020

How Copernicus found the periods and relative distances of the planets. Using provided data, students will be able to discover his methods and make their own calculations of planetary parameter using circles, angles and arcs, algebra formulas, and trig functions. Students also discuss data sources and opportunities for error. Oppportunities for teaching the lesson to different levels is discussed.
Pattern Busting
Authors: Dence Thomas
Published: November 2020

Even though sequences of numbers may follow an apparent pattern for a long period, that doesn't mean the pattern will continue forever. Geometric and algebraic sequences are used in examples as well as trig functions and prime number patterns.
Finding the Maximal Area of Bounded Polygons in a Circle
Authors: Rokach Arie
Published: November 2020

The goal of this article is to prove the following theorem by using only elementary geometric methods. The theorem is suitable for high school students and enables them to be actively involved in the proof process.
Internal Rate of Return
Authors: Shultz Harris S.
Published: November 2020

The Principles and Standards for School Mathematics (NCTM 2000, pp. 65–66) states, “School mathematics experiences at all levels should include opportunities to learn about mathematics by working on problems arising in contexts outside of mathematics. These connections can be to other subject areas and disciplines as well as to students' daily lives.” In this article we shall see that the discipline of finance can provide rich real–life applications of mathematics.
How Fair Is the Drug Test?
Authors: Lyublinskaya Irina E.
Published: November 2020

The non–medical use of anabolic/androgenic steroids among adolescents and young adults is of growing concern. As many as half a million Americans under age 18 may be abusing these drugs to improve athletic performance, appearance and self–image.
Tennis, Anyone?
Authors: Hodgson Ted R. , Burke Maurice J.
Published: November 2020

In this article, we present an engaging problem that is accessible to students at a variety of grade and skill levels. The problem is drawn from a common, real–world setting (tennis) and illustrates how a single problem can be solved in many ways by using increasingly powerful mathematics. We present these solution strategies as a sequence, beginning with informal hands–on activities and progressing to more formal and advanced mathematics. By considering the variety of solution strategies and by seeing how advanced mathematical techniques arise from basic properties and phenomena, students can develop a connected view of mathematics. The tennis problem allows students to develop an understanding of mathematical concepts and methods in a bounded setting.
Mathematical Lens: Hubcap display, Greenfield, Massachusetts
Authors: Turton Roger
Published: November 2020

A window display of hubcaps caught the eye of Roger Turton, who teaches in the mathematics department of the Stoneleigh-Burnham School in Greenfield, Massachusetts. Thanks to Turton for submitting the photograph and questions to “Mathematical Lens.”
Sharing Teaching Ideas: Did You Code?
Authors: Clausen Mary C.
Published: November 2020

If you were to observe one of my algebra classes on a day that the students were solving equations, you would probably hear me ask, “Did you code the equation?” Whether we are solving simple equations, multistep equations, multivariable equations, systems of equations using substitution, or radical equations, I have taught my students that the task will be easier if they “code” the equation.
Finding Complex Roots: Can You Trust Your Calculator?
Authors: Watson John W. , Ciesia Barbara A.
Published: November 2020

This article investigates a specific instance when the textbook answer for finding a root of a complex number differed with the answer given by the TI-83. After explaining the reason for the difference the article then expands the definition of the integral root of a complex number to an arbitrary complex power of a complex number. Read now to see where false assumptions might be made based on the results of a calculator and see explanations of how to overcome those assumptions with logic and proof.
ON THE EQUILIBRIUM OF A TRANSVERSELY ISOTROPIC ELASTIC HALF SPACE
Authors: HU HAI-CHANG  ,   中国科学院数学研究所
Published: November 2020

In a previous paper [1], we have obtained a general solution of three dimensional problem of the theory of elasticity for a transversely isotropic body. This general solution is applied in the present paper to the problem of equilibrium of a transversely isotropic half space. The following six problems are treated: 1) half space under given surface load, 2) half space under given surface displacement, 3) half space under given normal surface load and tangential surface displacement, 4) half space under given tangential surface load and normal surface displacement, 5) the contact problem of a rigid stamp upon a half space, 6) the problem of bending of thin plate resting upon a half space. It is found that, if, in problems 5) and 6), fricdonal force between stamp or plate and half space may be neglected, there exist simple relations between solutions for a transversely isotropic half space and corresponding solutions for an isotropic half space.
SPECTROCHEMICAL ANALYSIS OF COPPER IN IRON ORES
Authors: HO I-DJEN  , WANG THANG-SOO  ,   中国科学院金属研究所
Published: November 2020

A spectrographic procedure for determining copper in concentrated iron ores has been worked out for routine analysis. The powder method is found to be satisfactory with a medium quartz spectrograph and an activated a. c. arc between carbon electrodes as light source. In order to minimize erratic burning of the sample, the electrodes are pre-burned for one minute and then loaded with a mixture of equal weight of the ore sample and carbon powder. The analysis pair of lines chosen are: Cu 2824.369 Fe 2824.67 for copper contents ranging from 0.05 to 0.3%, and Cu 2824369- Fe 2828.813 for those between 0.3 and 1.0%. The results obtained spectrographically have been checked by wet chemical analysis, amongst 110 samples 80% of which the two methods checked within 0.02%.
ON THE SLOWING-DOWN OF NEUTRONS IN TWO ADJACENT MEDIA
Authors: HUANG TSXI-CHIA  ,   中国科学院物理研究所
Published: November 2020

We have calculated the slowing-down densities of neutrons in two adjacent media of different slowing-down properties according to the age theory. The outer medium is extended to infinity. The inner medium consists of a sphere of radius a, and contains a monoenergetic fast neutron-source in the form of a concentric spherical shell of radius R.The method of Laplace transformation is used in solving the problem. The results are given by the expansions (25), (25′), (28) & (28′), which converge rapidly for those cases we usually meet with.For a concrete case where the two media are heavy water and graphite, and the fast neutron-source is of energy 2 Mev, numerical results for the slowing-down densities at thermal energy are obtained and compared with those obtained by the usual two-group approximation (Fig. 1, 2 & 3).
Local hybrid optical encryption system based on double random phase encoding
Authors: Xu Xiang-Xin , Chang Jun , Wu Chu-Han , Song Da-Lin ,   School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China,   The First Research Institute of the Ministry of Public Security, Beijing 100044, China
Published: November 2020

Most of the existing selective encryption schemes are based on image processing and cannot be realized by optical structures, so their practicality is limited. Combining the optical design, a local hybrid optical encryption system based on double random phase encoding is proposed. The system proposed in this paper possesses a common aperture and dual optical path structure, which is widely used in optical design and can effectively improve the practicality of optical encryption system. First, important information and non-important information in the original image are separated by a selective beam splitter. Then light waves carrying important information enter into the 4f system for encryption, and light waves carrying non-important information enter into the diffraction system for encryption. Finally, part of the diffraction system ciphertext is replaced with 4f system ciphertext to obtain the final encrypted image. Decryption is the reverse process of encryption. First, the 4f system ciphertext is cut out from the final ciphertext. Then the 4f system ciphertext is used to restore the information replaced in the diffraction system ciphertext, thereby obtaining the complete diffraction system ciphertext. Finally, the two ciphertexts go through the reverse process of their respective systems to complete the decryption. By comparing the statistical characteristics and mean square error of the original image and the encrypted image, the effectiveness of the proposed system's encryption process is proved. By analyzing the peak signal-to-noise ratio of the original image and the decrypted image, the effectiveness of the proposed system's decryption process is proved. The sensitivity of each key of the system is analyzed to prove the security of the system. Especially the system is highly sensitive to selective encryption key, which proves the effectiveness and security of the proposed system for selective encryption. Through simulation, it is verified that the proposed system is very resistant to diffraction attacks. Even if he can obtain all the diffraction keys, the attacker still cannot obtain the selectively encrypted information. Finally, through simulation, it is verified that the proposed system has good noise resistance and crop resistance, and high robustness as well. The proposed system can realize the selective encryption through optical structure, which is safe, effective and highly robust, and thus improving the practicality of selective optical encryption system.