bluuebunny/crossref_metadata_2025 huggingface dataset:
Activities: Some Colorful Mathematics
Authors: DeTemple Duane W. , Walker Dean A.
Published: November 2020

In this article, we suggest three activities in discrete mathematics, all of which involve the coloration of geometric objects: counting colored regions of overlapping simple closed curves, counting colored triangulations of polygons, and determining the number of colors either necessary or sufficient to paint the points of the plane while satisfying a distance rule. They are designed to be used in small groups and can also serve as topics for whole-class discussions, offering opportunities for generating, collecting, and organizing data and examples; identifying, describing, and studying patterns; and making, testing, and verifying conjectures.
Connecting Research to Teaching: Professional Communities: Teachers Supporting Teachers
Authors: Adajian Lisa Byrd
Published: November 2020

In the past decade, numerous projects aimed at improving the teaching and learning of mathematics have been developed. One theme that has repeatedly emerged from these and other reform efforts is the importance of a strong professional community for helping teachers implement reform. When teachers are active participants in a professional community of their peers, they gain important knowledge and psychological support. In addition, when teachers' efforts are guided by their professional community, reform is more widespread and long lasting.
Technology Tips: Theorems in Motion: Using Dynamic Geometry to Gain Fresh Insights
Authors: Scher Daniel P.
Published: November 2020

Such software programs as The Geometer's Sketchpad (Jackiw 1995) and Cabri Geometry II (Texas Instruments 1994) allow students to turn the static images in their geometry textbooks into moving, deformable figures. With this ability comes the chance to pose challenges and approach theorems in ways that are not possible within a paper-and-pencil context. This article explores several dynamic-geometry constructions focusing on rectangles and shows how they foster fre h insights into traditional geometry theorems.
Technology Reviews, Publications and Products
Authors: Flanders Jim
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.
Sharing Teaching Ideas: A Function Is a Mail Carrier
Authors: Sand Mark
Published: November 2020

Most advanced mathematics is based on the concept of a function. Thus, it is imperative that students have a crystal-clear understanding of this fairly abstract idea.
Activities: Where Are We?
Authors: Waiveris Charles , Craine Timothy V.
Published: November 2020

Humans have been observing the motion of celestial bodies for thousands of years. This ancient knowledge coupled with the relatively recent introduction of latitude and longitude enables us to devise a simple procedure for determining our exact location on Earth. The mathematics for this activity is accessible to high school students and can be integrated with astronomy, geography, and history.
Implementing the Professional Standards for Teaching Mathematics: Developing as a Teacher of Mathematics
Authors: Bouck Mary , Keusch Teri , Fitzgerald William M.
Published: November 2020

A class of sixth-grade students had just finished working on a problem that challenged them to find decimal estimates for fractions. The students were using a set of fraction strips with whlch they were familiar. The teacher, Teri, shared the following analysis of her teaching of the lesson.
Projects
Authors: Stenglein Sharon
Published: November 2020

In 1989, the Geometry Learning Project (GLP) of the Curriculum Research and Development Group of the University of Hawaii set out to develop a high school geometry curriculum that effectively supports students' construction of geometric knowledge, carrying out the mandates of the NCTM's Standards documents (1989, 1991, 1995) and other calls for substantive change in the htgh school geometry curriculum. Following seven years of intensive research and field testing, which was funded by the National Science Foundation, the United States Department of Education, and the University of Hawaii, a final set of curriculum materials is being made available for broader dissemination.
College Preparation: What is its Effect on What you Teach and How You Teach It?
Authors: Wight Amelia C.
Published: November 2020

This old world has known many famous teachers. We like to gain inspiration for ourselves sometimes by thinking of them — of Socrates; of Plato, the pupil and intimate friend of Socrates; of Kant, who has been called the modern Socrates; of Dr. Arnold, the great headmaster of Rugby “to whom it was given many to save with himself”; of Edward Thring whose great axiom was “ The worse the material, the greater the skill of the worker”; and of many others whom I might mention and of whom you perhaps are thinking as I speak. We also like to think of the great army of men and women whose names we do not know, but who labored faithfully and untiringly in school-room and college-hall and who have quite as clear a title to the promise “ They that be teachers shall shine as the brightness of the firmament; and they that turn many to righteousness as the stars forever and ever.”
Should Arithmetic be Taught to all Pupils in the High School? When? How Much Time Should be Given to it?
Authors: Scobey Frank H. , Clapp Amy L. , Munhall Ruth
Published: November 2020

There is in my opinion no doubt whatever but that some arithmetic should be taught to all pupils in the high school.
Projects
Authors: Stenglein Sharon
Published: November 2020

Outreach and connections with K–12 education are part of the mission of the Geometry Center, a Science and Technology Research Center funded by the National Science Foundation and the University of Minnesota. One vehicle for this outreach is an intensive summer course for teachers in which they experience new learning using the center's technology resources. A graduate-level mathematics course, “Technology in the Geometry Classroom,” was created at the Geometry Center by director Richard McGehee with two graduate students, Eduardo Tabacman and Evelyn Sander, and two postdoctoral fellows, Chaim Goodman-Strauss and Heidi Burgiel.
Getting into the “Swing” of Functions
Authors: Clarke Lauren
Published: November 2020

Ever since Galileo Galilei pondered the swinging chandelier in the cathedral at Pisa, the pendulum has been a universal icon in physics and mathematics. Countless students have been introduced to the important concept of periodic motion through this simple device. Recent technology has given new life to the pendulum in the mathematics classroom.
Pascal's Triangle—Patte111s, Paths, and Plinko
Authors: Lemon Patricia
Published: November 2020

Iam currently teaching secondary mathematics in Kanata, Ontario. My Flllite Mathematics class comprises students who have completed our grade-11 mathematics course, which is roughly equivalent to two years of average-level high school algebra.
Reader Reflections
Authors: Goehl John F.
Published: November 2020

In a letter in the September 1996 Mathematics Teacher (“Problem 7, January 1996,” 501), Royce Huffman presents a variation of a Calendar problem noticed by his student, Buffy Last. The idea is that for certain two-digit numbers, their products are equal when their digits are reversed. For example, 69 × 64 = 96 × 46.
Reader Reflections
Authors: Almes Philip C.
Published: November 2020

The “Soundoffi” article in the November 1996 issue (“Who Is Dick Askey and Why Is He So Upset about the Standards?”) raises questions about the tendency of the Standards to stress applications over traditional topics. As one who as applied mathematics to various fields, I am pleased with extensions of the curriculum in that direction. Yet I also share the uneasiness about abandoning more traditional topics. Why not have both?
Implementing the Professional Standards for Teaching Mathematics: Honing the Abilities of the Mathematically Promising
Authors: Greenes Carole , Chapin Suzanne
Published: November 2020

As a consequence of concern that the Standards documents do not fully address the needs of gifted and talented students, the NCTM's Board of Directors appointed a task force in 1995 to consider curricular, pedagogical, and assessment issues related to the education of these students. Some of the questions raised in the 1996 report of the Task Force on the Mathematically Promising follow.
Parametric Equations, Maple, and Tubeplots
Authors: Feicht Louis
Published: November 2020

Parametric equations pose a problem for many students. Often students are unable to generate graphical representations quickly and find it difficult to interpret the “invisible” parameter. Such symbolic programs as Maple have parametric graphing capabilities built into them. Maple also has a form of graphing output called tubeplots, which requires the use of parametric functions. I have taught parametric graphing using tubeplots over the past four years. The mystery of parametric equations disappears with this hands-on approach as students create symbolic representations of functions from the graphical representations. This approach differs from the typical presentation of parametric equations in which students must create a graph from a symbolic representation. This activity establishes a graphical foundation for parametric equations and provides a comprehensive review and exploration of many previously learned topics.
Media Clips
Authors: Wood Dorothy
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 22091-1593.
Promote Systems of Linear Inequalities with Real-World Problems
Authors: Edwards Thomas G. , Chelst Kenneth R.
Published: November 2020

Because operations researchers solve problems in the real world, operations-research-based problems have rich connections to the world in which students live and work. Drawing on such problem situations is one way in which teachers can let applications of mathematics drive instruction. We believe that doing so will better motivate students to learn the mathematics they encounter in the classroom.
Extending a (TIMSS) Japanese Lesson Using Trigonometry
Authors: Pagni David L. , Shultz Harris S.
Published: November 2020

One of the Japanese mathematics lessons reported in the Third International Mathematics and Science Study (TIMSS) involves the concept of the area of a triangle. On the first day, students explore the area of triangles obtained by using the same base but translating the vertex opposite the base along a path parallel to the base, thus keeping the height constant. The next day the students are reminded of that property and are given a challenging problem that applies the property.
Activities for Students: Foam Images
Authors: Offerman Theresa Reardon
Published: November 2020

Consider this problem: The base of a solid is the region between one arch of y = sin x and the x-axis. Each cross section perpendicular to the x-axis is an equilateral triangle. Find the volume of the solid.
Experiencing Radians
Authors: Eggleton Patrick J.
Published: November 2020

As Einstein stated, most fundamental concepts in science, including mathematics, are essentially simple. Students do not have to memorize and us radian measure without understanding the concept. The simple correspondence between the radial angle and the measure of its arc to the measuring of fractional parts of a wheel in terms of the number of spokes furnishes a concrete representation of the concept of radians.
Developing Mathematical Reasoning Using Attribute Games
Authors: Quinn Anne Larson , Koca Robert M. , Weening Frederick
Published: November 2020

The game of Set has proved to be a very popular game at our college mathematics club meetings. Since we started playing, the membership has grown every month. In fact, one of our members brought her sixyear- old son to a meeting, and he now looks forward to playing Set with us. As a result of playing the game in our club and thinking about the results, we created and solved a variety of mathematical questions. For example, we wondered about possible strategies for winning and conjectured about phenomena that happened when playing. These questions involve a wide variety of traditional mathematical topics, including the multiplication principle, combinations and permutations, divisibility, modular arithmetic, and mathematical proof.
Service Learning: Taking Mathematics into the Real World
Authors: Duke Johnny I.
Published: November 2020

See “Sharing Teaching Ideas: When Will I Ever Use This Stuff, Anyway?” on page 798 for more information about service learning.
∑i=1ni2 and the Volume of a Cone
Authors: Greive Cedric E.
Published: November 2020

The sum of the first n squares is a series that has provided grist for the high school mill for decades (Hogben 1960; Rosenberg 1983; Hanisch 1984; Cooper 1986; Hegblom 1993). Here is yet another application of the series that will challenge even the best senior high school mathematicians-in-the-making: the derivation of the formula for the volume of a right circular cone using the expression for the sum of the first n squares combined with an algebraic solution of a limit as n approaches infinity.
Activites for Students: The Jurassic Classroom
Authors: Johnson Art
Published: November 2020

Proportional reasoning is a crucial part of any comprehensive mathematics program. Aspects of proportional reasoning can be introduced to students in primary grades, but most students are not developmentally ready to reason proportionally until upper middle school (Lawton 1993; Hart 1988; Piaget, Inhelder, and Szeminska 1960). A full understanding of proportionality opens the doors to many interesting applications of mathematics. A student who understands the proportionality involved in the areas and volumes of similar solids, for example, can understand why elephants and hippopotamuses have such short, massive legs; why a mouse can fall from a height of several stories and walk away, whereas a human will usually suffer fatal injuries; why insects cannot survive in extremely cold climates; why a bird or small mammal eats nearly its weight in a single day; and why a giant like Paul Bunyan could not exist. As early as 1638, Galileo wrestled with similar questions.
Consul, the Educated Monkey
Authors: Kolpas Sidney J. , Massion Gary R.
Published: November 2020

“Consul”, the Educated Monkey, is an outstanding, practical example of a plane linkage. In learning why the monkey works the way it does, students are required to review many important concepts from plane geometry, algebra, and arithmetic. Making their own “monkey” linkage similar to Consul, which one of the authors has done with construction paper and paper fasteners, would give students additional, hands–on experience with many important mathematical concepts.
Algebra for All: Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts
Authors: Leitze Annette , Kitt Nancy A.
Published: November 2020

Algebra for all really is possible. Unfortunately, too many students attempt to learn algebra by using memorization, and too many teachers use instructional methods that encourage memorization. Students struggle to learn algebra, and teachers struggle with how to best teach algebra so that it makes sense to their students and so that students will remember it beyond next week's test.
Projects: Building Regional Capacity
Authors: Kasten Peggy
Published: November 2020

Building Regional Capacity (BRC) is an NSF-funded professional development institute for grades 7—12 mathematics teachers, department heads, mathematics coordinators, and other present or future teacher leaders from around New England. Its chief focus is leadership in designing and delivering quality professional development. BRC was developed at the Education Development Center (EDC) and is a collaboration among EDC, the University of Massachusetts Lowell (UMass Lowell), and the Eastern Massachusetts Association of Mathematics Department Heads.
Teaching Strategies for Algebra for All
Authors: Choike James
Published: November 2020

Educators and policymakers alike recognize that algebra is an important gatekeeper course, not only for college preparation but also for preparation for the world of work. To prepare students for future success, many school districts and state legislatures now make algebra a graduation requirement for all high school students. Of course, the burden for preparing all students in algebra falls on the shoulders of the classroom teacher. Whether an algebrafor-all initiative is school-, district-, or state-based, a teacher faces the difficult challenge of teaching a high-standards course to a classroom of students whose beginning knowledge may range from far below to far above the course prerequisites.
Don't Teach Technology, Teach with Technology
Authors: Cyrus Vivian , Flora Ben V.
Published: November 2020

Graphing calculators and computers give students marvelous opportunities to explore and discover relationships more efficiently than they can by using nontechnological methods. However, lessons whose only objective is to learn about the technological tool are often not especially interesting and may not serve the best interests of the learner. In particular, we think that learning about the technology can occur better in the context of concept development. We believe that such learning not only encourages the use of the technology by supplying a real-world example but also enhances students' learning of the concept that is presented. The technology often furnishes opportunities for extending the original content of the lesson to include insights that could not have been addressed otherwise. An example is the following recent experience that we had with a model lesson presented to a group of high school teachers.
Projects: Delaware 6-12 Exemplary Mathematics Curriculum Implementation
Authors: Kasten Peggy
Published: November 2020

The Delaware 6–12 Exemplary Mathematics Curriculum Implementation (DEMCI) project seeks to substantially scale up from existing pilot projects to implement research-based, standardsdriven mathematics curricula in middle school and high school programs throughout Delaware. This National Science Foundation (NSF)–funded project is a partnership of fourteen Delaware school districts, the University of Delaware, and the Delaware Department of Education. Over the 38-month life of the project, 300 middle school and high school mathematics teachers—nearly two-thirds of all secondary mathematics teachers in Delaware— will engage in a substantial program of professional development that exceeds 150 hours for all teachers and may approach 200 hours for many.
Gloria Hewitt: Mathematician
Authors: Lattimore Randy
Published: November 2020

Experiences of African Americans in mathematics have been conspicuously absent from mainstream literature (Hilliard 1989).Gordon (1992, p. 27) points out the importance of incorporating minority life histories in education:
Activities for Students: The Archaeological Dig Site: Using Geometry to Reconstruct the Past
Authors: Moyer Patricia S. , Hsia Wei Shen
Published: November 2020

In secondary mathematics, students often see little connection between geometry and the real-life mathematical situations around them. When asked to describe geometric figures, their descriptions are sometimes no more than an identification of sides and angles. They have not had experience in using more than one property in a mathematical situation or in describing how two geometric properties are related. The van Hiele model of how students learn geometry proposes that students' understandings of geometry move from recognition to description to analysis (Fuys, Geddes, and Tischler 1988). For students to make this transition to analytic thinking, teachers need to create problem situations that enhance development of students' intuitive understandings. These investigations allow students to explore relationships among geometric shapes and to make conjectures about properties. The conjectures can then be stated formally as theorems.
Exploring the Birthday Paradox Using a Monte Carlo Simulation and Graphing Calculators
Authors: Whitney Matthew C.
Published: November 2020

A lesson using a graphing calculator and the “birthday problem” to stimulate student interest in probability.
Connecting Research to Teaching: Linking Task Characteristics to the Development of Symbol Sense
Authors: Kinzel Margaret Tatem
Published: November 2020

“Let x equal the number of miles.” How many explanations of solutions to word problems begin with a similar statement? As teachers, we frequently encourage students to state what their variables represent when they begin to solve a problem. However, we frequently see students who get lost in the notation and seem to have little or no means for finding their way out—or worse, students may not realize that they are lost. Lisa, a calculus student working on the Alpha and Beta travel task shown in figure 1, confidently began work on the task, used algebraic notation to represent quantities within the task, and arrived at a solution. In addition, Lisa was very comfortable with her answer. Lisa's work, as discussed in the next paragraph, was not correct and actually reveals a problem in her interpretation of notational expressions. We use this example to lead into a discussion of symbol sense, our goals for students, and how tasks can be used to help meet those goals. Readers may wish to try the tasks shown in figures 1, 3, 5, 7, and 9 before reading further.
Teaching Mathematical Induction: An Alternative Approach
Authors: Allen Lucas G.
Published: November 2020

An approach to teaching mathematical induction that builds on student knowledge of the recursive form of a function.
Technology Tips: October 2001
Authors: Starnes Daren , Gosse Paul
Published: November 2020

Al Coons kicks off this month's column with an excellent demonstration of how he uses grouping to organize and store data, regression models, and graphs. Our second tip gives a brief introduction to TI InterActive!, the dynamic mathematics software from Texas Instruments. Not surprisingly, we chose to feature the TI InterActive! Web site for this month's installment of “Surfing notes.”
Stand Up and Be Counted: The Mathematics of Congressional Apportionment
Authors: House Peggy A.
Published: November 2020

Since the founding of the United States, the principal reason for the decennial census has been to determine the allotment of seats in the House of Representatives.
Functions of Number Theory in Music
Authors: Johnson Craig J.
Published: November 2020

Connecting mathematics with other disciplines is one of the major goals of many general-education mathematics courses taught at universities and colleges across the nation. Of course, from kindergarten to calculus, good teachers demonstrate and explore these connections as a means of making clear the relevance and importance of mathematics to every person. In particular, I have cheerfully noticed a renewed interest in the relationships between mathematics and music. This article examines several fundamental notions of number theory as they arise in music theory, and in the process, develops several music-related functions. I see an added pedagogical benefit in emphasizing the importance of functions in mathematics with examples from a different arena than the usual function pool. Middle school and high school teachers might find that a subset of these topics are handy nuggets of material that can show examples of applications of mathematics outside the mainstream uses in science and engineering. I created this article as notes for the liberal-arts mathematics course that I teach at my college and have consistently found this topic to be one of the students' favorites.
Mathematics Didn't Just Happen!
Authors: Lightner James E.
Published: November 2020

Students of the history of mathematics have long enjoyed the fascinating stories of the mathematical discoveries that have occurred over the past 4000 or so years. In considering the early developments, that is, those that happened before A.D. 100, we must often conjecture about who made the discovery or observation, where it really took place, and exactly when it happened. We still do not always know why the developments occurred when they did. However, mathematical developments often seem to be tied to the history of the world; certain periods of time have certainly been much more conducive to major mathematical activity than others. In other words, mathematics did not just happen; it was encouraged—or discouraged— by the zeitgeist—the nature of the prevailing culture— and the political, social, and economic conditions under which mathematicians had to work.
Connecting Research to Teaching: When Visualization Is a Barrier to Mathematical Understanding
Authors: Aspinwall Leslie , Shaw Kenneth L.
Published: November 2020

AS MATHEMATICS TEACHERS, OUR INTUITION TELLS us that students benefit from being able to understand a variety of representations for mathematical concepts and being able to select and apply a representation that is suited to a particular mathematical task. Students develop mathematical power as they learn to operate on mathematical objects and to translate from one mathematical representation to another when necessary. For example, graphic representations convey mathematical information visually, whereas expressions represented symbolically may be easier to manipulate, analyze, or transform. The National Council of Teachers of Mathematics (NCTM) reinforces our intuition: “Different representations support different ways of thinking about and manipulating mathematical objects. An object can be better understood when viewed through multiple lenses” (NCTM 2000, p. 360). Notwithstanding our intuition and experience with students' representations, we teachers tend to think that students' graphic and visual abilities are always an advantage in mathematics learning. Indeed, we accept on pedagogical faith that students' conceptual understandings are enhanced whenever they use visualization.
Reader Reflections: February 2003
Authors: Gladbach Jennifer
Published: November 2020

I assigned a p project for 3/14/02. The following poem is by Jennifer Weingarten, and it exceeded my expectations.
Using a Simple Optical Rangefinder to Teach Similar Triangles
Authors: Cuicchi Paul M. , Hutchison Paul S.
Published: November 2020

Simple range-finding and height-finding devices that use the properties of similar triangles are common. For example, foresters use these devices to measure the heights of trees, and some golfers use them to measure the distance from ball to hole, although such devices are illegal in competitive golf. Simple optical range-finding golf scopes are inexpensive and are reasonably accurate.
Penny Packing for Your Thoughts
Authors: Bezuszka Stanley J. , Kenney Margaret J.
Published: November 2020

Packing pennies onto a piece of paper provides perplexing problems for students to ponder. To set the stage for students to gain insight into penny-packing problems, we suggest that you provide groups of students with small sheets of paper and an ample supply of pennies. At the start of the investigation, a quarter of a standard-size 8.5 in.- by-11 in. sheet of paper is sufficient for each group. Ask the groups to cover their paper completely with pennies. Students typically choose one of the covering patterns shown in figure 1 and figure 2. As part of the task, students should determine the number of pennies needed to completely cover the paper using each pattern. The penny-packing problem is, Which pattern guarantees the most pennies on the page?
The Play-off Probability Problem
Authors: Lauber Murray L.
Published: November 2020

few years ago, just as I was about to introduce binomial probabilities in my precalculus class, the Edmonton Oilers were in a first-round play-off series with the Dallas Stars. Each team had won a game. The series suggested a problem: given that the Oilers had a probability p of winning any game, what was the probability that they would win the series? I focus on the Oilers because the small university where I teach is located a one-hour drive from their home in Edmonton. Our initial figure of p = .3 was based loosely on the Oilers' record against the Stars. We began with what I will call the brute-force method, treating the rest of the series as a five-game series. After completing the brute-force solution, we searched for a shorter, more elegant, solution. Although the solutions that we unearthed along our path of discovery are not new, they illustrate beautifully the process by which many mathematical problems are solved, extended, and generalized.
Mathematics, Politics, and Greenhouse Gas Intensity: An Example of Using Pólya's Problem-Solving Strategy
Authors: Yoshinobu Stan T.
Published: November 2020

While reading the newspaper one morning in spring 2002, I came across an article (Shogren 2002) about the Clear Skies and Global Climate Change initiatives introduced by the White House. The initiatives propose to cut greenhouse gas intensity by 18 percent over the next ten years (Clear Skies and Global Climate Change Initiatives 2002). My first reaction was to think, “What in the world is greenhouse gas intensity?” The term greenhouse gas intensity is not defined in the article, so I had no way to be sure what it measures. Anyhow, reducing greenhouse gas intensity should be a step in the right direction for our planet. The scientific evidence overwhelmingly indicates that the earth is warmer than its historical average.
Technology Tips: February 2004
Authors: Hollebrands Karen Flanagan , Stohl Hollylynne
Published: November 2020

This month's “Technology Tips” introduces readers to a powerful teacher-productivity tool. Mathematics teachers have needed inexpensive, easy-to-use software that allows them to create electronic documents that contain multiple representations. One such software package is TI-Interactive! (version 1.1, Texas Instruments, 2002). TI-Interactive is a word-processing tool that combines the features of a graphing calculator with the flexibility of a word-processing program, an equation editor, and a computer algebra system. This software tool gives mathematics teachers the power to create, modify, and share textbook-quality graphics and symbols. This “Technology Tip,” written by Robin L. Rider, is meant to introduce some of the commonly used features of the software. Future “Technology Tips” will explore more advanced features.
Fairness: How to Achieve It and How to Optimize in a Fair-Division Procedure
Authors: Jones Michael A. , Cohen Stanley F.
Published: November 2020

Many disputes all over the world–on international, national, local, and personal levels–involve the division of items for which two parties have equal claim. We describe a simple yet practical application of mathematics for fairly resolving disputes. It requires no more mathematical sophistication than algebra. But what does “fairly” mean? And what does it have to do with mathematics? We explain the basic properties of fair divisions and show how to determine mathematically whether these properties are met. We introduce the adjusted-winner procedure of dividing disputed goods between two parties or players by dividing the prizes won in a fictitious contest. We also consider how one player can use information about his or her opponent's preferences for the disputed goods to maximize his or her return under the adjusted-winner procedure. Through this exercise, students learn concepts about optimization, without introducing calculus.
The Constant Feature: Spanning K–12 Mathematics
Authors: Schultz James E.
Published: November 2020

When I reflect on my many years of teaching mathematics, one of the most enjoyable and fruitful ideas for calculator use came from an article that appeared more than twenty-five years ago (Judd 1976). It gave suggestions for using the constant feature of a simple calculator to enhance mathematics learning. I have used Judd's idea successfully in many classrooms at grades levels ranging from kindergarten through twelfth grade. This article revisits the constant feature for four-function calculators and extends it for use with graphing calculators and computer algebra systems (CASs) for a broad range of topics.
Focusing on Students' Mathematical Thinking
Authors: Breyfogle M. Lynn , Herbel-Eisenmann Beth A.
Published: November 2020

When viewing videotaped examples of his classroom teaching, Anthony, a veteran ninth-grade teacher, was surprised that he focused more on the students' responses than on the students' thinking. For example, he realized that he was not asking questions to understand what or how the students were thinking but rather to test their knowledge.
How Many Votes Are Needed to Be Elected President?
Authors: Mahoney John F.
Published: November 2020

The winner of the popular vote is not necessarily the candidate who is elected president. The paper describes a process, using mathematical modeling, by which students can try to determine how few votes are necessary in order to be elected president. Students use ratio and proportion to make predictions and make comparisons to 1888 data.
Farey Sums and Understanding Ratios
Authors: Mihaila Ioana
Published: November 2020

Farey sums are explored as a way to solve mixture, distance, and sequence problems. Authors examine the proper way to add ratios and give application problems.
From the Editors: Patterns: Revitalizing Recurring Themes in School Mathematics
Authors: Sandefur Jim , Camp Dane
Published: November 2020

Patterns are the very essence of mathematics, the language in which it is expressed. In recognizing and creating patterns of every conceivable type, mathematics is an art. It is also the science of analyzing and synthesizing such patterns. The art and science of exploring patterns have been entwined with mathematics from the very beginning, a relationship that is akin to the role of rhyme in poetry.
Activities for Students: Exponential Growth through Pattern Exploration
Authors: Brown Betty
Published: November 2020

Do your students think of arithmetic, algebra, and geometry as separate entities? Do the laws of exponents tend to confuse them? Participants at the summer institute Pattern Exploration: Integrating Math and Science in the Middle Grades developed and used the three activities in this article. The activities have proved to be exceptional tools for providing students in middle school and the early years of high school with models of exponent laws while relating differing fields of mathematics.
Delving Deeper: Multiplying Everything Using the Difference of Two Squares
Authors: Reinstein David
Published: November 2020

Great music and stories can take us to unexpected places and bring us great joy in the process. So can numbers. In our high school courses, we often skim surfaces that cover fascinating mathematics. This article is about my exploration of the difference of two squares formula and how it relates to prime numbers. You may want to use a calculator or pencil and paper as you read this so that you can play along.
How Do We Know That's the Minimum?
Authors: Franzblau Deborah S.
Published: November 2020

Reasoning mathematically is a habit of mind, and like all habits, it must be developed through consistent use in many contexts.
Delving Deeper: Viewing Standard Problems in Nonstandard Ways
Authors: Haznedar Asena , Hurwitz Marsha , Fromhold Albert T.
Published: November 2020

In this month's column, we hear from two teachers and a student who discover different approaches to ordinary problems.
Sharing Teaching Ideas: Solving Absolute Value Equations Algebraically and Geometrically
Authors: Wei Shiyuan (Steve)
Published: November 2020

Understanding the geometrical meaning of algebraic equations or solving algebraic equations geometrically can help students improve their comprehension.
Non-Geometry Mathematics and The Geometer's Sketchpad
Authors: Moyer Todd O.
Published: November 2020

The Geometer's Sketchpad (GSP) is a well-known interactive geometry software package. Its usefulness in geometry instruction has been well researched (Choi-Koh 1999; Dixon 1997; Groman 1996; Lester 1996; Moyer 2003; Weaver and Quinn 1999). Finzer and Jackiw (1998) recommend the use of GSP as the dynamic manipulative for geometric concepts. GSP allows students to construct a figure, to perform measurements of lengths and angles, and then to “click and drag” any part or parts of that figure to look for change.
Rotations of the Regular Polyhedra
Authors: Jones MaryClara , Soto-Johnson Hortensia
Published: November 2020

The mathematical sciences particularly exhibit order, symmetry, and limitations; and these are the greatest forms of the beautiful.
Changes in the physico-chemical properties of topsoil in a landslide-affected area (western part of the Transylvanian Basin, Romania)
Authors: BĂLC Ramona , ROBA Carmen , ROSIAN Gheorghe , COSTIN Dan , HORVATH Csaba , ZGLOBIU Octavia Raluca , CHIRTOS Diana
Published: November 2020

Landslides determine increases and decreases in specific soil compounds which is affecting soil fertility. The recovery of soil fertility is a long process and may be used as an indicator of the landslide age and can contribute to the management plan of the affected area. In order to add to data about soil properties affected by landslides, the current study focuses on a young and shallow landslide from the western part of the Transylvanian Depression. Soil samples were analysed from a physico-chemical point of view (pH, organic matter – OM, total organic carbon – TOC, major cations, and iron content) in two places, at between 0 and 60 cm depth (inside and outside the landslide). The results obtained showed lower values of pH inside the landslide, low values of TOC and rock fragments in both places studied (inside and outside the landslide) and no differences in soil texture between disturbed and undisturbed soil. The ammonium, magnesium and calcium content was higher outside the landslide, the sodium level was slightly higher outside the landslide, while the potassium concentration was higher inside the landslide. This study offers new data regarding recovery of soil fertility and highlights the importance of gaining knowledge on soil properties of relevance to future measures to increase the fertility of agricultural soils.
TWO TYPES OF UNDER-WATER QUARTZ TRANSDUCERS
Authors: HO SHOW-AN  , CHOW SUEI-SHIEN  ,   中国科学院应用物理研究所
Published: November 2020

Two types of under-water quartz transducers, namely, the mosaic type & the array type, were designed and manufactured, particularly paying attention to the points of cementing of crystals and supporting of vibrating bodies, with a view to reducing the equivalent resistance r of the transducers when they are unloaded. A transducer of the mosaic type with an area of 609 mm2, unsupported, measured 60 k Ω for its r, while one of the array type with an area of 259 mm2, 15 k Ω. Finally, discussions were made on the comparison of these two types of transducers with the conclusion that a transducer of the array type possesses more merits than that of the mosaic type. The ways for further improvements were considered.
GENERAL EQUATIONS OF SANDWICH PLATES UNDER TRANSVERSE LOADS AND EDGEWISE SHEARS AND COMPRESSIONS
Authors: TU CHING-HUA  ,   清华大学
Published: November 2020

Non-linear differential equations have been developed for the calculation of the deflections and buckling loads of Sandwich plates subjected to transverse loads and edgewise shears and compressions.The Sandwich plate consists of two thin metal face sheets bonded to a light slab-the core. The core is taken to have zero moduli as deformations in its own plane. This appears to give an adequate representation of behavior of many core materials.Some previous theories are shown to be special cases of this more general theory. The essence of non-linearity are shown through the dicussian of core properties and action of interface-and-core forces. The restriction of some special forms of displacements of practical interest has been studied and the existence of nonlinear element in the bending problem is clarified. The relative merit of using large bending deflection theory instead of the small bending deflection theory in the faces is given.
ON THE GAS FLOW AND HEAT TRANSFER IN LAMINAR BOUNDARY LAYER FLOW
Authors: LIN HUNG-SUN  ,   中国科学院数学研究所
Published: November 2020

The equations of the plane, laminar compressible boundary layer flow past a plane or slightly curved wall
ON THE BENDING OF RECTANGULAR PLATES WITH CLAMPED EDGES
Authors: HU HAI-CHANG  ,   中国科学院数学研究所
Published: November 2020

The problem of bending of orthotropic rectangular plates with clamped edges on elastic foundation may be reduced to the following differential equation and boundary conditions (?4w)/(?x4)+2λ(?4w)/(?x2?y2)+(?4w)/(?y4)+kw=q/D. w=0, (?w)/(?x)=0 at x=±a, w=0, (?w)/(?y)=0, at y=±b. In the case of isotropic plates, λ = 1. In this paper a perturbation method is proposed for the solution of this problem fay expanding w in power series of λ: w=w0+w1λ+w2λ2+……. It is proved that this series is convergent when -1 ≤λ≤1.
INTERNAL FRICTION PEAKS ASSOCIATED WITH THE STRESS-INDUCED DIFFUSION OF CARBON IN FACECENTERED CUBIC ALLOY-STEELS AND METALS
Authors: KE TING-SUI  , WANG CHI-MEN  ,   中国科学院金属研究所
Published: November 2020

Internal friction peaks asscciated with the presence of carbon in several types of f.c.c. alloy-steel (18/8 type stainless steel and high manganese steel) have been observed from measurements with a torsion pendulum. The temperature for maximum internal friction lies between 200-300℃with a frequency of vibration of about I cycle per second. The height uf the peak rises and the position of the peak shifts to a lower temperature with an increase of the carbon content. When the amount of carbon in solid solution is reduced by tempering the specimen at an elevated temperature, the height of the peak lowers and the peak shifts to a higher temperature. A comparison of the activation energy and the diffusion ccefficients determined by internal friction methods with those measured in conventional macro-diffusion experiments reveals that the observed internal friction peak is associated with the stress-indused diffusion of carbon in these face-centered cubic steels.Internal friction peaks associated with the stress-induced diffusion of carbon in a Ni-Al alloy and in pure nickel have also been observed. These experiments show that the appearance of an internal friction peak associated with the diffusion of carbon in f.c.c. metals may be a general phenomenon.
RADIUS OF NUCLEAR CHARGE DISTRIBUTION AND NUCLEAR BINDING ENERGY
Authors: TSENG CHIN-YUEN  ,   北京大学物理系
Published: November 2020

From the analysis of the experimental results (high energy electron scattering, X-ray spectra of μ-mesonic atoms), it is pointed out that the radius of nuclear charge distribution, Rp, is very nearly proportional to the cubic root of the atomic number z1/3, ins tead of the usually accepted relation Rp∝A1/3. Under this assumption, we have modified the semi-empirical mass formula. The Coulomb energy term is now assummed to be of the form 3/5 (z2e2)/ropz1/3).The aagreement of the nuclear binding energies calculated fromthe new fomula with the experimental data is definitely better than that of the old Bethe-Weizsacker formula. Also the mass numbers of the most beta-stable nuclei are very correctly predicated by the new formula.
CALCULATION OF THE COHESIVE ENERGY OF METALS BY THE FERMI-THOMAS METHOD
Authors: CHENG KAI-CHIA  ,   南京大学
Published: November 2020

Slater and Krutter have attempted to solve the problem of cohesive energy of metals by using the Fermi-Thomas method. They treated the valence electrons and the electrons in the ion core on the same footing. In a way similar to Seitz's cellular method, a sphere with a volume equal to that of a unit cell was drawn and periodic boundary conditions were applied on the surface. Furthermore another boundary condition was imposed in their calculation, namely, on the ground that the sphere as a whole is electrically neutral, the potential energy of an electron on the surface should be equal to zero. Then it was expected that there existed a minimum of the total energy versus the interatomic distance. However, actual computation gave no such a minimum at all. This discrepancy was ascribed to the neglect of the correlation energy of the electrons.It now appears to the present author that the failure to find a minimum is the result of an incorrect boundary condition imposed by the previous authors. In view of the fact that each electron is only acted by all the other electrons except the one being considered, obviously an electron on the surface on the average receives a potential energy-e2/R. Using this boundary condition we can calculate the total energy by numerical integration of the equation, in which the data of Feynman, Metropolis and Teller are adopted. A minimum of the energy versus the interatomic distances is found. The calculated interatomic distances for the minimum for various elements together with the observed ones are tabulated and compared. For the multivalent heavy elements the agreement is remarkably good. For the monovalent light elements, e.g. for the alkali metals, the agreement is not so satisfactory, giving only an order of magnitude. This, however, is expected from the statistical nature of the present method, which certainly is not adquate for the description of the single valence electrons, e.g. those in the alkali metals.For the cohesive energy of the metals, the total energy of the system is subtracted by the energy of the free atoms. We use the Fermi-Thomas model for the free atoms also. In view of the fact that for the free atoms the Fermi-Thomas model is accurate only within 1%, the result of the subtraction would introduce an error which is about hundred times that of the total cohesive energy. Hence the absolute value of the cohesive energy up to the present stage is not reliable. The author believes that this method can be extended to the calculation of the energy of formation of alloys, where the total energy of a free atom is not necessarily known. It is hoped that this programm will be carried out in the near future.
ON THE DECOMPOSITION OF QUENCHING GAS IN G-M COUNTERS
Authors: LI TEH-PING  , KIU SHIH-HUA  ,   中国科学院物理研究所
Published: November 2020

Measurements of the pressure-rise in G-M counters filled with argon and alcohol indicate that, corresponding to each ion pair produced in the discharge, the increase in number of gaseous molecules due to alcohol decomposition, lies between 5 to 10. This number increases as the overvoltage is lowered. Thus the counting life of a counter can not be expressed merely in terms of quantity of electric charge passing through it.Some possible explanations are discussed.