bluuebunny/crossref_metadata_2025 huggingface dataset:
Independence of the Incidence Postulates
Authors: Huffman David C.
Published: January 2021

CONSISTENCY is an essential property of any mathematical system. Without it two conflicting statements may be deduced within the system. Independence, on the other hand, is not considered essential, and, indeed, it is sometimes not desirable. If the set of postulates in a high school geometry text were independent, then the first few basic theorems would be so difficult that virtually no high school student could understand their proofs. At the foundations level, however, it is considered aesthetically desirable that the postulates of a system be independent. It is generally not an easy matter to prove that a set of postulates is independent.1 It is believed that when Euclid stated his five postulates for geometry he thought that his parallel postulate was demonstrable from the other postulates, but he was unable to give a valid proof of it. Mathematicians sought a proof for two thousand years, but it has been only within the last century or so that the search has been shown to be futile. It took two thousand years and the development of non-Euclidean geometries to show that the parallel postulate is independent.
The Hazards of Sets
Authors: Geddes Dorothy , Lipsey Sally I.
Published: January 2021

The use of the set concept promises clarification, simplification, and unification in our teaching of mathematics.1 Many points in our presentation of mathmatics can be brought into better focus through the use of sets, i.e., fundamental laws of algebra, arithmetic operations, definitions such as solutions of equations and inequalities, variables, functions and inverse functions, congruence, line segments and curves, and the notion of classification. By means of sets we can more effectively distinguish between angle (a set of points) and measure of an angle (a number) and among various geometric configurations. Simplification in teaching many mathematical ideas is achieved by building on the youngsters' intuitive notion of set, which has been or can be developed through physical and visual exposure to sets.2
Have You Read…?
Authors: Peak Philip
Published: January 2021

Prime numbers have fascinated man for years; they are the source of many questions and have led to the identification of many important concepts. This article gives some interesting ideas about primes, such as: there are an infinite number of palindromic primes such as 313, 929, 73,637; there are, however, also “reversibles” which are not palindromic. For example, there are 93 five-digit palindromes such as 35153 but there are 1,366 other reversibles that are prime but are not palindromic. Palindromes and reversibles must begin and end with 1, 3, 7, or 9. Palindromes cannot have an even number of digits, but this is not true of reversibles. A unique six-digit prime is 193939, which is reversible and cyclic. The six numbers formed by successively moving the initial digit to the end are each reversible. Similarly, 11939 is unique among five-digit primes. There are no such numbers with four or seven digits.
Historically Speaking— The Parallel Postulate
Authors: Eves Howard , Rolwing Raymond H. , Levine Maita
Published: January 2021

Euclid's famous parallel postulate was responsible for an enormons amount of mathematical activity over a period of more than twenty centuries. The failure of mathematicians to prove Euclid's statement, from his other postulates contributed to Euclid's fame and eventually led to the invention of non-Euclidean geometries.
“McKay's Theorem” and Farey Fractions
Authors: Kriewall Thomas E.
Published: January 2021

“McKay's Theorem” is the title of a recent article in the Mathematics Teacher (Sberzer 1973). It reports an incident of sudden insight by a student named McKay and the reaction of his teacher, Laurence Sherzer. In brief, Sherzer reports that he nearly snatched defeat from the jaws of victory when he was tempted to say, “No, McKay, that's not how it's done.”
Memorizing Trick
Authors: Beasley Guy L.
Published: January 2021

Many teachers of mathematics find they can create interest and a desire to learn by using entertaining devices or “trick” problems not found in the textbook. Some of these can be easily turned into a learning situation. One such device that I find to be very useful goes as follows.
Let’s Use Trigonometry
Authors: Rodgers John J.
Published: January 2021

All too often in the teaching of high school mathematics courses, we overlook the inherent flexibility and interdependence of the subject matter. It is easy to fall into the trap of presenting algebra, trigonometry, geometry, and so on, as separate areas of study. It is because they were taught this way traditionally. With relatively minor changes in the order of presentation, we can demonstrate to the student the vital interconnectiveness of mathematics. For example, many courses in high school geometry include a unit on trigonometry. The student learns three trigonometric ratios, namely, the sine, the cosine, and the tangent. He also learns to use the trigonometric tables to solve for an unknown side of a right triangle. Generally this material comes quite late in the year.
Rallying with Mathematics
Authors: Lindsey Gaye P.
Published: January 2021

There are innumerable ways to disguise drill problems for junior high school students—using cross-number puzzles, bingo games, connect-the-dot pictures, and so on. However, older students usually find these methods childish and boring. In an effort to provide “fun” drills for senior high school algebra students, I drew on my experience in sports car rallies and made up a road map rally. It could be adapted to other classes simply by varying the problems involved.
More on Repeating Decimals
Authors: Jacobs Neal
Published: January 2021

This paper continues the discussion of repeating decimals taken up in this journal by Margaret M. Hilferty (1972) and Margaret R. Hutchinson (1972). It presents a table that can be used to predict the length of the repeating part of the decimal representation of the reduced fraction k/n for any natural number n all of whose prime factors are less than 1,000.
The Use of Vectors in Proving Classical Geometric Theorems
Authors: White Paul A.
Published: January 2021

In the November 1972 issue of this journal, Thomas W. Shilgalis showed how to use transformation geometry to prove the classical theorem that says that the centroid G, orthometer O, and circumcenter H of a triangle are collinear. Furthermore, the distance from G to O is twice that from G to H. This result also gives a beautiful example of the use of vectors in proving geometric facts. With the increasing occurrence of linear algebra as a mathematical tool in many disciplines, it is important to include the fundamentals of the subject as early as possible. It is also important to have real applications, and geometry provides an abundance of them.
Search for a New Field
Authors: Arnold David H.
Published: January 2021

The field of complex numbers can be an exciting new topic for students to explore. A class can help discover some motivation for that inscrutable definition for multiplying complex numbers. The trend now seems to be for authors to introduce complex numbers as ordered pairs of real numbers, but the potential for any student participation in this exploration is usually quickly destroyed by the introduction of that mysterious, difficult, and totally unmotivated rule for multiplying these new numbers. It takes the students a while to memorize a definition that they do not understand. But then checking the field properties is as easy as it is boring! What can teachers do to avoid this trap? Why not let the class help in the pursuit of an acceptable definition for the operation of multiplication on these ordered pairs?
Pie in the Street or How to Calculate π from the License Plates in the Parking Lot
Authors: Vervoort Gerardus
Published: January 2021

An interesting view of π. Students should enjoy this refreshing approach to the computation of approximations to π.
Do Similar Figures Always Have the Same Shape?
Authors: Kumpel Paul G.
Published: January 2021

Here's transformational geometry applied to conics with a challenge to extend the idea to graphs of cubics.
Congruence Extended: A Setting for Activity in Geometry
Authors: Spitler Gail , Weinstein Marian
Published: January 2021

One way to add relevance and interest to a geometry course is to provide opportunities for students to conjecture about possible theorems, to test these conjectures inductively, and then to prove the theorems where appropriate. The following is a brief outline of a unit on congruence of geometric figures for the secondary level (either junior high or high school).
New Programs
Authors: Montague Margariete A. , Odom Mary Margaret , Tobin Alexander
Published: January 2021

As the NCTM was publishing its 1972 yearbook. The Slow Learner in Mathematics, teachers at Glenbrook South High School, Glenview. Illinois, were engaged in the design of a curriculum for slow learners in all major academic subjects. After a study of relevant literature, the teachers determined that their efforts should be interdisciplinary. The Core Learning Project (CORE) reflects this decision. Two freshmen courses were prepared—Communication Arts (English, Social Studies, Art and Music) and MATHSCI 153 (Mathematics and Science). The subject integration was continued at the sophomore level with American Studies and MATHSCI 253.
Minimal Surfaces Rediscovered
Authors: Ehrmann Sister Rita M.
Published: January 2021

History, exploration, imagination, discovery—all these are nicely blended in looking at some of the familiar geometric configurations.
A Classification of Projectile Paths
Authors: Opp Roger L.
Published: January 2021

Here is an application in analytic geometry.
Trigonometry Revisited: A Combined Approach
Authors: Satty Glenn Jay
Published: January 2021

The method of developing trigonometry described here combines several typical approaches. It aims for minimum memorization but maximum insight into mathematical thought.
Events in the History of American Mathematics: 1876 ± 100
Authors: Jones Phillip S.
Published: January 2021

This bicentennial year naturally turns our thoughts back to colonial times, back to vignettes of 1776. However, there are two reasons for including a longer span in a historical note. First, history is both more interesting and more valuable when it analyzes forces and issues operating over a longer interval of time. Second, there was relatively little of mathematical or pedagogical importance going on in the America of 1776.
GCD, LCM, and Boolean Algebra?
Authors: Cohen Martin P. , Juraschek William A.
Published: January 2021

The familiar process of finding the GCD and LCM can be treated as binary operations with interesting consequences.
The Harmonic Mean: A Nomograph, and Some Problems
Authors: Skidell Akiva
Published: January 2021

A familiar mathematical operation in several fascinating situations.
The “Piling Up of Squares” in Ancient China
Authors: Swetz Frank
Published: January 2021

About the same time Euclid was compiling the Elements, on the other side of the globe the brilliant work of Chinese scholars was being collected in an ancient text. The solution technique illustrated here, which was developed to a high art in China, should intrigue you and your students.
A Discovery Activity in Geometry
Authors: Mathews John H. , Leonard William A.
Published: January 2021

Are you tired of the same geometric proofs? Take a look at this activity.
Some Novel Consequences of the Midline Theorem
Authors: Hoehn Larry
Published: January 2021

Geometry students will enjoy exploring some new uses of a familiar theorem.
Almost Congruent Triangles With Integral Sides
Authors: Briggs John T. F.
Published: January 2021

Here is a good enrichment activity for your geometry students.
Least Squares and the 3:40-Minute Mile
Authors: Oakley Cletus O. , Baker Justine C.
Published: January 2021

Can we predict when, if ever, a 3:40 mile will be run?
What Can be Done With a Mira?
Authors: Lott Johnny W. , Dayoub Iris Mack
Published: January 2021

Have you and your students discovered this powerful tool for the study of reflections and the Euclidean constructions?
A Minimathematical Problem: The Magic Triangles of Yates
Authors: Carmony Lowell A.
Published: January 2021

The corner numbers are very significant in the solution of these magic triangles.
A Pentamino Unit
Authors: Ferrell Phyllis C.
Published: January 2021

You can win at pentamino if you watch your shapes.
Identifying and Teaching Essential Mathematical Skills—Items
Authors: Cowan Richard E. , Clary Robert C.
Published: January 2021

A thorough process for identifying mathematical skills is described, and the skills are listed.
Sharing Teaching Ideas
Authors: Nicolai Michael B.
Published: January 2021

Sharing Teaching Ideas offers practical tips on the teaching of topics related to the secondary school curriculum. We hope to include classroom-tested approaches that offer new slants on familiar subjects for the beginning and the experienced teacher. Please send an original and four copies of your ideas to the managing editor for review.
People, People, People
Authors: Dunn Samuel L. , Chamberlain Ruth , Ashby Patricia , Christensen Kenneth
Published: January 2021

Provocative questions and guidelines for classroom investigation of population growth.
Finding Areas Under Curves with Hand-Held Calculators
Authors: Hiatt Arthur A.
Published: January 2021

No, this is not the usual limit of the sum from calculus. All you need are some points on the curve—and, of course, your calculator.
A General Test of Divisibility
Authors: Smith Lehi T.
Published: January 2021

A sequel to the divisibility story of December 1976
A School Computer Network
Authors: McCaig Duncan A. , Jansson Lars C.
Published: January 2021

Four considerations in introducing computers into secondary schools are (1) the cost, (2) the type of hardware, (3) the software, and (4) the curriculum. Clearly the four are interdependent, although the nature of their relationships has not been comprehensively articulated. Although curricular goals should perhaps be the starting point in educational planning of this kind, the present paper will focus on one attempt to coordinate software and hardware for an existing or predefined curriculum at a reasonable cost.
Odd Pie Fights
Authors: Carmony Lowell
Published: January 2021

Here are same surprising results and even a chance far induction with cream pies.
A Novel Way to Factor Quadratic Polynomials
Authors: Autrey Melanie Ann , Austin Joe Dan
Published: January 2021

A classroom-tested alternative for factoring. Give it a try.
Rectangular Solids with Integral Sides
Authors: Prielipp Robert W. , Oman John A. , Kuenzi Norbert J.
Published: January 2021

What happens if the sides of a rectangular solid have lengths with no common factor greater than I?
Cube Roots on A Calculator— Some More Thoughts
Authors: Peterson Gregory K.
Published: January 2021

Cube roots revisited with an infinite series and implications for trisection.
Postscript on A Visit to The People's Republic of China
Authors: Swetz Frank
Published: January 2021

The balancing of social relevance, rigor, and quality is a critical task for Chinese mathematics educators.
Implementing Mathematics Proficiency Testing
Authors: Mott Warren
Published: January 2021

Due to public demand for increased emphasis on basic skills, the California Legislature revised the education code. Effective 1 January 1977 the new regulations require school districts to establish proficiency standards in reading, writing, and computation, and to develop procedures to assess student proficiency in the above areas for grades four through twelve. After June 1980, any student who has not met the locally adopted standards will not be given a diploma.
Roman Numeral Puzzle
Authors: Spieler Robert
Published: January 2021

Find the corresponding arabic numerals for the following roman numerals and enter your results in the appropriate places in the puzzle.
The Time to Organize a National Mathematics League is Now
Authors: Kalfus Alfred
Published: January 2021

Our students want it. Our students need it. Our students deserve it. And our profession will grow because of it. Let's do it now!
Phonograph Records and Adding Machine Paper
Authors: Hirstein James J.
Published: January 2021

Spiraling down while going ‘round and ‘round… an attack on a perplexing problem of length.
Geometric Growth and The Hand-Held Calculator
Authors: Sloyer C. W.
Published: January 2021

A secondary school teacher that I worked with told me of his method of introducing geometric growth (geometric series) in the secondary classroom.
Mathematics in Photography
Authors: Jeffrey Neil J.
Published: January 2021

One look at the markings on an adjustable camera demonstrates that photography depends heavily on mathematics. On such a camera are marked f-stops, shutter speeds, and distance and ASA scales. In the camera manual are exposure tables and depth-of-field charts. Packed with each roll of film are development times and temperatures. And in the literature are innumerable charts and graphs. In this paper I will attempt to explain some of the mathematics involved in photography.
Algebra and The Computer
Authors: Landry Michael
Published: January 2021

A story of how one teacher devised a technique for solving linear equations. Can you think of other applications in elementary algebra?
The Current Status of Computer Literacy: NAEP Results for Secondary Students
Authors: Carpenter Thomas P. , Corbitt Mary Kay , Kepner Henry S. , Lindquist Mary Montgomery , Reys Robert E.
Published: January 2021

Computers have become an integral part of our society, as seen for example, in twenty-four-hour banks, computerized bills and paychecks, computer-assisted learning, computerized medical diagnosis and care, and computerized appliances. Since such diverse applications of computers are already commonplace and promise to increase in the future, it is important that our educational programs inform and prepare students with respect to the potential capability of computer systems as well as for the direct implications they have on our lives.
Apportionment-A Decennial Problem
Authors: Sullivan John J.
Published: January 2021

Arithmetic and algebra can help your students understand some of the dilemmas in devising a plan for equitable representation in Washington.
Line Fitting Using Only High School Algebra
Authors: Staib John
Published: January 2021

Experimentally obtained data never fall on a line. Here’s how to find the equation of a Une that is close to most of the points.
Making Change: a Problemsolving Activity
Authors: Levine Deborah R.
Published: January 2021

When you paid for that 200 candy bar yesterday, did you take two dimes out of your pocket; four nickels; or perhaps one dime, one nickel, and five pennies? In that choice lies the beginning of a journey into pattern finding, generalizing, and deductive thinking that your students may not have traveled before.
Green Globs: a Microcomputer Application for Graphing of Equations
Authors: Dugdale Sharon
Published: January 2021

Computers have an unprecedented ability to do tedious calculations almost instantly. leaving the student able to attend to larger concepts. This ability, coupled with the growing availability of low-cost computers with sophisticated display capabilities, makes computers a natural choice for materials intended to improve students’ understanding of functions and graphs. This article outlines the development of an activity that uses the computer's unique capabilities to provide students with a relevant and highly motivating experience with the graphing of equations.
Formal Axiomatic Systems and Computer-Generated Theorems
Authors: Battista Michael T.
Published: January 2021

One of the main topics of discussion in Douglas Hofstadter’s Pulitzer-prize-win-ning book Godei, Escher, Bach: An Eternal Golden Braid is the concept of deriving theorems in a formal system. To give his readers a better idea of what a formal system is, Hofstadter discusses a simple example of such a system, called the MIU system. Although the example may not have been designed for instructional use, to a teacher of mathematics it seems ready-made for the classroom. Indeed, the MIU system can not only provide an instructional unit on some important topics in mathematics, it can also be used to lead students to investigate some fascinating ideas from computer science and artificial intelligence. The present article will describe the MIU system and indicate a number of possible uses for it in the mathematics classroom.
Some Quick Constructions
Authors: Waters William M.
Published: January 2021

The story is told (Eves, p. 216) of the German mathematician Adam Riese (1492-1559) winning a contest with a draftsman. The challenge was to construct as many lines perpendicular to each other as possible during one minute, using only the compass and an unmarked ruler—that is, by classical Euclidean methods. The draftsman proceeded to construct his figures in the usual way, ending up with a set similar to that in figure I.
The Multiplication of Signed Numbers
Authors: Zlot William , Roberts Ralph E.
Published: January 2021

The basis of the following approach to signed numbers lies in the fundamental geometric concept known as a similarity transformation. The presentation relies on ordinary magnifying lenses, which are, themselves, worthy of study. We will discuss problems involving the multiplying of integers; however, the approach applies as well to the multiplying of any signed numbers. Experience indicates that teacher demonstration followed by hands-on activity in small groups and the use of suitable diagrams and verbal descriptions constitute a successful classroom presentation.
Activities: Finding Factors Physically
Authors: Hirsch Christian R.
Published: January 2021

Materials: Scissors, red pens or markers, and a set of the activity sheets for each student. Cutouts from two transparencies of sheet 4 would be helpful for discussing student solutions to the problems.
Soma®: A Unique Object For Mathematical Study
Authors: Spector David
Published: January 2021

The Soma® puzzle is a 3 × 3 × 3 cube composed of seven different pieces (see fig. 1). Many people are familiar with Soma® simply as a puzzle in which the object is to construct a cube. Few realize that the Soma* puzzle can be used in counting and construction activities with the average student. It can be equally valuable when used to provide enrichment and higher-level activities for the accelerated student. While manipulating the cubes, students make use of critical thinking skills: analysis, investigatory skills, conceptualizing, and reasoning skills in a context of spatial relations. The following activities have been used with gifted elementary students for two years as part of my work with a Title IV project, “Inter- age Program for Critical Thinking.”
Mathematical Excitement—The Most Effective Motivation
Authors: Long Calvin T.
Published: January 2021

Have you never seen a lovely pattern, a beautiful theorem, or an elegant proof. I feel sure that you have and that it was these very things that first attracted your interest and caused you to pursue your studies in mathematics. As Henri Poincaré (Stewart 1952, p. 187), the French mathematician and philosopher, once remarked, “Every scientist worthy of the name, but above all a mathematician, experiences in his work the same sensations as an artist; his pleasures are as great and of the same nature.” And Karl Weier-strass (Smith 1947, p. 1), the great German analyst, once wrote that “a mathematician who is not somewhat of a poet will never be a perfect mathematician.” In the same vein. W. F. White (Stewart 1952. p. 162) aptly penned the following: “The beautiful has its place in mathematics for here are triumphs of the creative imagination, beautiful theorems, proofs, and processes whose perfection of form has made them classic. He must be a ‘practical’ man who can see no poetry in mathematics.”
Probabilites In Yahtzee
Authors: Litwiller Bonnie H. , Duncan David R.
Published: January 2021

Teachers of units in probability are often interested in providing examples of probabilistic situations in a nonclassroom setting. Games are a rich source of such probabilities. Many people enjoy playing a commercial game called Yahtzee. A Yahtzee player receives points for achieving various specified numerical combinations of five dice during the three rolls that constitute a turn.
Adding Dimension to Flatland: A Novel Approach to Geometry
Authors: Esbenshade Donald H.
Published: January 2021

“Why do we have to take geometry?” This question has provoked an ongoing discussion among mathematics teachers and students about what to include in the high school geometry course (Hoffer 1981; Norris 1981). In this context I have tried to do something to help expand my students’ perceptions of geometry.
Activities: Finding Patterns in Linear Relations
Authors: Wallace Edward C.
Published: January 2021

This section is designed to provide mathematical activities suitable for reproduction in worksheet and transparency form for classroom use. This material may be photoreproduced by classroom teachers for use in their own classes without requesting permission from the National Council of Teachers of Mathematics. Laboratory experiences, discovery activities, and model constructions drawn from the topics of seventh, eighth, and ninth grades are most welcome for review.
Sharing Teaching Ideas: The Cartesian Classroom
Authors: Avery Jean
Published: January 2021

Sharing Teaching Ideas offers practical lips on the teaching of topics related to the secondary school curriculum. We hope to include classroom-tested approaches that offer new slants on familiar subjects for the beginning and the experienced teacher. Please send an original and four copies of your ideas to the managing editor for review.
Microcomputer-assisted discoveries: Primes
Authors: Kimberling Clark
Published: January 2021

Microcomputers can be used by students to make their own discoveries. Those who know BASIC or are just learning it, and those who have an interest in prime numbers, will want to use the programs in this article as a basis for their own computer-assisted discoveries. Although these programs are written in Applesoft BASIC II, they are easily adapted to other microcomputer dialects of BASIC.
What Are These Things Called Variables?
Authors: Wagner Sigrid
Published: January 2021

We all know some of the problems that beginning algebra students have in working with literal symbols, or variables. Two articles in this journal (Herscovics and Kieran 1980; Rosnick 1981) have discussed certain difficulties related to variables and have offered suggestions for overcoming them.
SoundOFF: The Buying and Selling of Textbooks— Friend or Foe of the Curriculum?
Authors: Denman Theresa I.
Published: January 2021

“The largest single group of unprotected consumers is made up of the 50 million schoolchildren now required to use thousands of inadequately evaluated educational materials.” This revelation voiced in 1971 (Komoski; 1971, p. 9) by the then president of Educational Products Information Exchange (EPIE) further stressed that an analysis by the exchange of sixty “best selling” textbooks disclosed that fewer than 10 percent had ever been field-tested prior to publication.
Microcomputer-assisted discoveries: Generate Your Own Random Numbers
Authors: Kimberling Clark
Published: January 2021

“Many random-number generators in use today are not very good,” wrote Donald Knuth in The Art of Computer Programming (1961). But this is only one of several reasons for having students try to generate their own random numbers.
Honeycomb Geometry: Applied Mathematics in Nature
Authors: Roberts William J.
Published: January 2021

“Why do the bees make such funny looking holes?” asked one inquisitive child during a recent visit to our apiary. When I looked at the question seriously, it was clear that the reason for the hexagonal shape could be understood by a high school class in geometry and that exploring its application would be interesting toward the end of the semester.
Composing “Interesting” Exercises Involving Rational Expressions
Authors: Hornsby E. John , Cole Jeffery A.
Published: January 2021

Under what conditions will a solution strategy to a rational equation include inadmissible values as possible solutions? The authors had a spirited discussion about this question one afternoon. We hope to offer some insight into methods of generating such equations and to provide some generalizations that can be used when writing classroom examples or test items. One can also apply the techniques to writing sums of rational expressions that can be reduced to lower terms once the initial sum has been obtained. The algebraic techniques required are not beyond the capabilities of gifted secondary school students.
Advanced Placement Computer Science
Authors: Braswell James S.
Published: January 2021

The College Board's Advanced Placement (AP) offering in computer science is in its first year of operation. In the spring of 1983 the board (1984) published a course description to serve as a guide to those secondary schools that wish to offer AP Computer Science. This course description is also the basis of the first AP examination in computer science that is being administered in May 1984. A teacher's guide for AP Computer Science (College Board 1983) has also been prepared to assist secondary school teachers in planning and teaching the course.
SoundOFF: The Computer—A Tool for Instruction?
Authors: Wilkinson Jack D.
Published: January 2021

“January of 1980 is an important month in the history of computers and people. During that month, more computers were built than children were born.” This statement by Arthur Luehrmann should set the stage for a September 1984 statement that describes the significant, positive impact that computers have had on secondary school mathematics education (grades 7-12). However, no such statement can be made.
Activities: Plotting and Predicting from Pairs
Authors: Shulte Albert P. , Swift Jim
Published: January 2021

The collection of data, the selection of appropriate displays that illuminate the data, and the interpretation of those displays are all basic skills in today's information-oriented world.
Microcomputer-assisted discoveries: Lines
Authors: Kimberling Clark
Published: January 2021

“Microcomputer-assisted Discoveries” offers tips for using the microcomputer in novel ways to improve the teaching of mathematics. Please send five copies of your ideas to the managing editor for review.
SoundOFF: We Can Dream, Can’T We?
Authors: Seymour Dale
Published: January 2021

The views expressed in the “Soundoff” editorials do not necessarily reflect the views of the Editorial Panel of the Mathematics Teacher or the National Council of Teachers of Mathematics. Readers are encouraged to react to these editorials by writing to the author with copies to the Mathematics Teacher for consideration in “Reader Reflections.” Please double-space all letters that are to be considered for publication. Editorials from readers are welcome.
A Curious Number Pattern
Authors: Landauer Edwin G.
Published: January 2021

In everyday life, mathematical curiosities often arise that can be used in the classroom to study the techniques of problem solving. These curiosities offer an interesting alternative to the traditional methods of attacking problems. Usually, our students are studying one type of problem, and therefore, the method of solving that problem becomes routine. When a problem arises from a source outside the classroom, however, a student may have to investigate several techniques before arriving at the appropriate procedure for a solution.
Sharing teaching ideas: An Algebraic Puzzler
Authors: Lulli Henry
Published: January 2021

“Sharing Teaching Ideas” offers practical tips on the teaching of topics related to the secondary school curriculum. We hope to include classroom-tested approaches that offer new slants on familiar subjects for the beginning and the experienced teacher. See the masthead page for details on submitting manuscripts for review.
The “Difference” in Babbage's Difference Engine
Authors: Crowley Mary L.
Published: January 2021

Have you ever wondered how mathematical tables were developed, who was interested in producing them, and what kinds of calculations were involved? Recently, I found myself asking these questions while reading a biography of Charles Babbage (1791 1871). Mathematician, scientist, economist, and inventor, Babbage was an important figure in nineteenth-century English society.
High-Resolution Plots of Trigonometric Functions
Authors: Stick Marvin E. , Stick Michael J.
Published: January 2021

At an early level, most students become familiar with the traditional twodimensional plots using X-Y rectangular coordinates. Given a series of data points in the X-Y plane, these same students should normally have little difficulty displaying their data given the appropriate scales with which to work. The more adventuresome computer buffs will probably delight in generating various BASIC plot routines on a microcomputer.
A Brief Historical Dictionary of Mathematical Terms
Authors: Mcginty Robert , Mutch William , Beynen John Van
Published: January 2021

I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history.
Sharing Teaching Ideas
Authors: Camp Dane R. , Woods John M. , Orzechowski Larry A.
Published: January 2021

“Sharing Teaching Ideas” offers practical tips on the teaching of topics related to the secondary school curriculum. We hope to include classroom-tested approaches that offer new slants on familiar subjects for the beginning and the experienced teacher. See the masthead page for details on submitting manuscripts for review.
Soundoff: Computers and Mathematics—An Unholy Alliance?
Authors: Luehrmann Arthur
Published: January 2021

Ask your fellow citizens what a computer is, and guess what you will hear. “A giant calculator,” “A number cruncher.” “A machine that does math very fast.” And so on. In the minds of most people, the modern computer is tied closely to the world of arithmetic. Yet most electronic computers hardly do arithmetic at all. About a dozen years ago, a study at a major scientific research laboratory aimed at finding out what fraction of the time its giant “number cruncher” actually did arithmetic. Only 5 percent of the time involved adding, subtracting, multiplying, and dividing numbers. The other 95 percent involved the moving of information from place to place, testing for logical truth, and counting. In actual practice today, a computer is an information machine, not an arithmetic machine.
Expanding the Limits of the Calculator Display
Authors: Sherzer Laurence
Published: January 2021

The decimal expansions of 1/7 or 5/13 can be found immediately on the average electronic calculator. These decimal expansions are limited to six digits before they begin to repeat. But what do we say to students who want to explore the repeating properties of rational numbers? What would we answer if they asked for the decimal expansion of 1/17 or 3/23? Do we just say, “Keep dividing and you'll find the answer”? Hardly. Long decimal expansions are usually left to actuaries who don't want to mismanage thousands of dollars over the life of an annuity or mortgage table, or to spacecraft engineers who don't want to miss their targets in outer space.
New Questions about Numbers Outside the Mathematics Classroom
Authors: Smart James R.
Published: January 2021

The greater-than-expected success of the original article, “Questions about Numbers Outside the Mathematics Classroom” (Mathematics Teacher 72 [November 1979]:624–25), has resulted in repeated requests for a second test of the same sort. These real-life uses of numbers on the new test have been common over a long period of time, yet many of us have not stopped to consider them carefully.
An Interesting Solid
Authors: Shahin Louis
Published: January 2021

Some plane figures are interesting because their perimeter and area are numerically equal. For example, a square with a side of 4 cm has a perimeter and area numerically equal to sixteen. A further study of plane figures with this interesting property can be found in Bates (1979) and Markowitz (1981).
The Geometric Supposer: Promoting Thinking and Learning
Authors: Yerushalmy Michal , Houde Richard A.
Published: January 2021

Traditionally, the teaching of high school geometry has emphasized the principles of deductive systems. This approach often forces students to learn how to manipulate mathematical systems while it denies them an equal opportunity to create geometry. Geometry teachers have always faced the dilemma of having to instil in their students an appreciation of deductive mathematical systems while at the same time offering them an opportunity to create mathematics. This article describes our approach in dealing with this dilemma.
The Bank Shot
Authors: Byrne Dan
Published: January 2021

During a game of pool, a player often discovers that no direct shot into a pocket is possible. When confronted with this predicament, the player will often resort to a bank shot. Figure 1 illustrates a typical situation in which the dark object ball is to be banked into the side pocket.
When Is a Program Like a Function?
Authors: Klotz Frederic S.
Published: January 2021

In some respects computer programming seems closely related to mathematics. Good programming requires the same combination of clear logic, attention to detail, and inventiveness that is characteristic of good mathematics. Yet many important concepts in programming—loops, pointers, arrays, and floating-point arithmetic, to name just a few are awkward to deal with in mathematical terms. Even the concept of a variable in computer science looks strange to someone accustomed to variables in mathematics.
The Twelve Days of Christmas and the Number of Diagonals in a Polygon
Authors: McMaster Adrian
Published: January 2021

What an absurd title! Does some common element link the song “The Twelve Days of Christmas” to the number of diagonals in a polygon?
Complexity of Geometric k-Planarity for Fixed k
Authors: Schaefer Marcus
Published: January 2021

The rectilinear local crossing number, $\mathop{\overline{\rm lcr}}(G)$, of a graph $G$ is the smallest $k$ so that $G$ has a straight-line drawing with at most $k$ crossings along each edge. We show that deciding whether $\mathop{\overline{\rm lcr}}(G) \leq k$ for a fixed $k$ is complete for the existential theory of the reals, $\exists \mathbb{R}$.
Ponencias de la Jornada UX 2020 “Interactions for Care”
Authors: Mosaic
Published: January 2021

El pasado mes de noviembre se celebró la XI edición de la Jornada de Experiencia de Usuario de 2020 (JUX Mosaic 2020), enmarcada en los estudios de multimedia, videojuegos y UX de la Universitat Oberta de Catalunya (UOC).
Bajo el título “Interactions for Care”, el tema central de este año fue el diseño de experiencias e interacciones para el cuidado, la salud y el bienestar de las personas.
Comparison as a Lens: Interpretation of the Cultural Script of a Korean Mathematics Lesson Through the Perspective of International Lesson Study
Authors: Sarkar Arani Mohammad Reza , Shibata Yoshiaki , Cheon Ho-seong , Sakamoto Masanobu , Kuno Hiroyuki
Published: January 2021

This study aims to examine how cross cultural analysis can lead to deeper understanding of the cultural script of teaching, and how teachers learn to transform their teaching script through a research-based transnational learning platform. In this study, emphasis is placed on a cross
cultural analysis to view in depth the cultural script of teaching mathematics in Korea through the eyes of Japanese teachers and critical lenses of researchers’ feedback. The objective herein is to focus on the challenges whereby Korean teachers have to redesign teaching as they look
at students as problem solvers. This cross cultural analysis attempts to determine the cultural script of teaching mathematics in Korea and improve the quality of teaching from the following two perspectives: 1) teacher teaching, and 2) student learning.
Real-time change of optic nerve sheath diameter after rebleeding of ruptured intracranial dissecting aneurysm
Authors: Koo Hae-Won
Published: January 2021

Ultrasonographic measurement of optic nerve sheath diameter (ONSD) has been validated to detect and monitor the increased intracranial pressure (IICP) in neurosurgical field. Especially, the ONSD has been known to reflect well the ICP in case of intracranial hemorrhage (ICH) occurring in the anterior circulation of intracranial vascular system, however it has not been well elucidated the role of ONSD in the posterior circulating vascular problems. A 43-year-old man presented with a subarachnoid hemorrhage with intraventricular hemorrhage due to rupture of dissecting aneurysm at right vertebral artery, and immediately performed the stent-assisted coil embolization. Two day after coil embolization, he became nearly alert without definite neurologic deficits. The ONSD was reduced from 5.8 mm to 5.5 mm. The 10th postoperative days, the patient suddenly changed into stuporous mentality due to rebleeding of aneurysm. The ONSD was enlarged to 6.7 mm. We report a case of intimate relationship between ONSD and IICP after rupture of dissecting aneurysm located in posterior circulation.
COVID-19 pandemic – mode of transmission and preventative measures to stop its spreading
Authors: Parrey Imtiyaz Rasool
Published: January 2021

The purpose of our research was into  providing a protocol to prevent the spread of SARS-CoV-2 infection in light of the limited information related to this coronavirus. In detail, we broke down and looked focused on proof based rules gave in the different nations influenced by this pestilence up till now. In addition, we broke down the suggestions for the counteraction and control of different pandemics brought about by different pathogens having a place with a similar group of coronaviruses or others that present similar instruments of transmission Coronavirus Disease 2019 (COVID-19) have emerged, and given the momentum nonappearance of exceptionally powerful endorsed antibodies or medications, beast power approaches including physical hindrances are being utilized to counter infection spread. A significant reason for physical insurance from respiratory contaminations is eye, nose, and mouth security. Nonetheless, eye insurance with goggles is dangerous due to "misting", while nose/mouth assurance is confused by the breathing challenges related with non-valved respirators.Key words:  Coronavirus, Pandemic, Transmission, Preventative Measures.
Unraveling the cooperative effects of acid sites and kinetics for pyrolysis of CHF3 to C2F4 and C3F6 on SO42−/ZrO2‐SiO2
Authors: Wang Gang , Cai Guangming
Published: January 2021

The treatment or upgrading of waste trifluoromethane (CHF3, R23), which has a significant greenhouse effect, is of great importance in industry. Herein, series of SO42−/ZrO2‐SiO2 catalysts with different Brønsted and Lewis acid site densities and ratios were prepared for pyrolysis of R23 to tetrafluoroethylene (C2F4, TFE) and hexafluoropropylene (C3F6, HFP). The effects of impregnation concentration of (NH4)2SO4 on specific surface area, crystal phase, and Brønsted and Lewis acid site densities and ratios were respectively demonstrated. The Brønsted and Lewis acid sites were observed to have cooperative effects on R23 transformation and up to 94.6% selectivity of (TFE + HFP) could be achieved at 750°C. The kinetic studies revealed the decomposition of R23 into CF2 carbene and HF was the rate‐determining step, and a deactivation behavior was found due to the site coverage and pore blockage by the oligomers of TFE and HFP.
Synthesis, characterization and evaluation of novel ferrocenylmethylamine derivatives as cytotoxic agents
Authors: Savani Chirag J. , Roy Hetal , Verma Sanjay K. , Vennapu Dushyanth R. , Singh Vinay K.
Published: January 2021

The present report describes a new series of amide functionalized 20‐ and 30‐aminomethylferrocene derived from ferrocenylmethylamine. The compounds 1a‐5a and 1b‐5b were characterized by microanalysis, 1H, 13C NMR, UV–visible, fluorescence, FTIR, thermogravimetric and crystallographic techniques. The X‐ray analysis demonstrated the ability of these molecules to form various intermolecular hydrogen bonding interactions, as verified by Hirshfeld surface analysis. All the compounds were evaluated against MCF 7, IMR 32, HepG2 and immortal L132 cell lines by MTT assay and the results were compared with cisplatin. Interestingly, many compounds were very active against all the investigated cell lines and proved to be more potent as cytotoxic agents than cisplatin. The western blot, gene expression, mitochondrial membrane potential and flow cytometry study were used to investigate the mode of action of these derivatives as antitumor agents. The results showed apoptotic property of the compounds by modulating inflammatory pathway against human tumor cells of different origin. We performed the density functional theory calculations and molecular docking to rationalize the experimental results.
Highly Stable Plating/Stripping Behavior of Zinc Metal Anodes in Aqueous Zinc Batteries Regulated by Quaternary Ammonium Cationic Salts
Authors: Qiu Qiliang , Chi Xiaowei , Huang Jiaqi , Du Yuexiu , Liu Yu
Published: January 2021

Aqueous zinc‐ion batteries (AZIBs) have attracted wide attention as large‐scale energy storage systems, owing to their obvious merits such as high safety, low cost, and simple maintenance. However, the development of AZIBs is seriously hindered by Zn dendrite‐induced low efficiency, cell fading, and even cell short‐circuiting. Unlike the traditional anode modification strategies, this work stabilized the Zn anode by functionalizing the aqueous electrolyte with a series of quaternary ammonium cationic compounds. The cations of the compounds can be adsorbed on the surface of the metallic Zn during electrodeposition process, which alters the nucleation and deposition behavior of zinc ions according to the molecular structure of the cations. Tetraethyl ammonium bromide (TEAB) was, for the first time, demonstrated to guide homogeneous Zn deposition, thereby inhibiting the growth of dendrites in near‐neutral electrolytes. Highly reversible Zn stripping/plating performance with an average coulombic efficiency of 99.7 % is achieved. Particularly, the Zn−Zn symmetric cell based on TEAB‐containing electrolyte exhibits an ultra‐long cycling time of 3000 h at 1 mA cm−2 and even 800 h at 5 mA cm−2; the Zn−MnO2 full cell shows stable cycling under the condition of the practical‐level mass loading.
Channel estimation based on low‐complexity hierarchical codebook design for millimeter‐wave MIMO systems
Authors: Hadji Baghdad , Fergani Lamya , Djeddou Mustapha
Published: January 2021

Millimeter‐wave bands (mmWave) are considered as a strong candidate for achieving high‐quality communication links for the future outdoor cellular systems to overcome the spectrum congestion problem. Due to the extremely high path loss in mmWave band, large antenna arrays at both the transmitter and receiver are necessary. Hybrid beamforming architectures are used to exploit the potential array gain with several RF chains, which poses a problem of complexity when estimating the mmWave channel. To address the challenge of this hardware complexity, we propose in this paper an approach to design a multiresolution hierarchical codebook to meet parallel multistream data based on the physical design via hybrid analog/digital architecture with low complexity, i.e., 2‐bit phase state and few numbers of RF chains. The simulation results verify that our proposed method to design the codebook has better design performance of beams and can achieve higher average spectral efficiency gains of channel estimation compared to the one based on high hardware complexity.