bluuebunny/crossref_metadata_2025 huggingface dataset:
Word Roots in Geometry
Authors: McIntosh Margaret E.
Published: November 2020

Vocabulary and word study are now considered to be within the purview of the mathematics classroom. But, as mathematics teachers, we rarely do a separate unit on “words.” Instead, we teach the words Students' learning logs helped me gauge their tnderstanding 510 our students need—triangles, quadratic equations, and so on—along with the regular unit of study. This article offers suggestions for a unit on word study in geometry. This unit lasts one week and is best presented as the first unit of the year.
Fibonacci Meets the TI-82
Authors: Dion Gloria S.
Published: November 2020

Graphing calculators have had a significant impact at many levels in high school and college mathematics in the past few years. We have had to reexamine what we teach and how we teach. Topics that were once the exclusive province of calculus, such as max-min problems, are currently studied by algebra students working with graphing calculators. We have added to our syllabi new techniques, such as zoom and trace; new concepts, such as complete graph and friendly viewing window; and new mathematics, such as chaos and fractals. Graphing calculators have helped us enhance, enrich, and enliven our courses, and they have empowered students, allowing them to discover and explore mathematics as active learners.
Linear Functions with Two Points of Intersection?
Authors: Contino Michael A.
Published: November 2020

Of course two straight lines in Euclidean space cannot intersect in more than one point unless they are the same line and intersect everywhere—or can they? Follow this problem on the graphing calculator, however, and the surprising twist that gives this article its name will be seen. The material covered should be readily accessible to first-year-algebra students who have studied systems of equations, but it also contains valuable lessons for college mathematics professors who have been easily deceived by its apparent simplicity and familiarity.
Technology Tips: From Drawing to Construction with The Geometer's Sketchpad
Authors: Finzer William F. , Bennett Dan S.
Published: November 2020

Students have many reasons for making a sketch with The Geometer“s Sketchpad. Their purpose may be to explore the behavior of a particular geometric figure, such as a rhombus, or to model a New users must understand the difference between a drawing and a construction physical situation, such as a ladder leaning against a wall. They may want to make a beautiful pattern inspired by Navajo rug designs, or their goal may be an animation—perhaps a Ferris wheel or a merry-go-round. No matter what the purpose, they want the finished sketch to behave well when they drag basic elements around in it. If investigating properties of rhombuses, they want the figure to remain a rhombus while they drag to explore its different sizes and shapes. Students want to end up with a construction, not merely a drawing.
Geometry Proof Writing: A Problem-Solving Approach à la Pólya
Authors: McGivney Jean M. , DeFranco Thomas C.
Published: November 2020

The Third Committee on Geometry, composed of twenty-six prominent teachers in the field of mathematics, prepared a questionnaire which raised pertinent questions concerning the teaching of geometry. The replies indicate that “there is almost unanimous agreement that demonstrative geometry can be so taught that it will develop the power to reason logical ly more readily than other school subjects, and that the degree of transfer of this logical training to situations outside geometry is a fair measure of the efficacy of the instruction. However great the partisan bias in this expression of opinion, the question ‘Do teachers of geometry ordinarily teach in such a way as to secure transfer of those methods, attitudes, and appreciations which are commonly said to be most easily transferable?’ elicits an almost unanimous but sorrowful ‘No.’” (Fawcett 1938, 8)
Technology Tips: Visualizing the Behavior of Functions
Authors: Cuoco Albert A.
Published: November 2020

Tought experiments is a technique I often have stuents perform in my classes. When working on a problem that asks for the rectangle of a given perimeter that maximizes area, I ask students to imagine a long skinny rectangle gradually changing into a tall skinny rectangle. The image in my head is a continuously changing rectangle that starts as something close to a horizontal segment of length half the perimeter and gradually gains height and approaches a vertical segment. I would say that about half the students can do such an experiment easily. For the other half, this visualization takes a great deal of work. Chalk is not very helpful in gettmg them to make the right images; videotapes are better, but students all too easily become passive spectators when watching a television set.
Counting Tile Patterns
Authors: Grassl Richard
Published: November 2020

Would your students be surprised to see a use for partial fractions? Would they benefit from “jumping in” and constructing some mathematics, even though they may have had little prior experience with collecting data, working in groups, and “talking mathematics”? Students who have taken part in this exploration—which involves drawing pictures to collect data, conjecturing a recurrence relation, and solving such a relation using geometric series and partial fractions—have been genuinely pleased to see the pieces fit together and their cooperative effort yield a signficant final product. The activity develops connections among mathematical topics, as described in the NCTM’s Curriculum and Evaluation Standards for School Mathematics (1989), and promotes teaching discrete mathematics at the secondary school level. It has been classroom tested at several levels, including a sophomore college-level course in discrete mathematics for prospective elementary and secondary school teachers.
Writing: A Necessary Tool for Learning
Authors: Elliott Wanda Leigh
Published: November 2020

As the author's recertification deadline approached, a search began for a class that would meet the necessary requirements. A colleague suggested a writing class, which aroused both fear and curiosity. While free-writing during the first class meeting, this author realized that the act of writing stimulates thought. Students who are given a focused free-writing assignment must think about the material so they can write about a concept and define it in their own words. By the end of that first class, the author discerned that writing to learn would change her teaching forever. She came to believe that writing is a necessary tool for learning mathematics.
Technology Tips: Make These Designs
Authors: Picciotto Henri
Published: November 2020

Make These Designs” is an activity that I have used in algebra classes over many years. It can be done with just about any electronic grapher. Its purpose is to reinforce students' understanding of the connection between the graph of a linear function and the parameters m and b in the formula y = mx + b.
Soundoff!–A Computer for All Students—Revisited
Authors: Waits Bert K. , Demana Franklin
Published: November 2020

Ten years ago graphing calculators started a revolution in the teaching and learning of mathematics. In early 1992 we published an article in the Mathematics Teacher describing inexpensive graphing calculators as computers with built-in graphing software (Demana and Waits 1992). These calculators could be viewed as computers available to all students because of their low cost, ease of use, and portability. Our point was that only a few elite would benefit if teachers had to rely exclusively on expensive computer laboratories to deliver computer enhanced visualization in mathematics teaching and learning.
When Does a Dog Become Older Than Its Owner?
Authors: Quinn Anne Larson , Larson Karen R.
Published: November 2020

The relationship between algebra and arithmetic is not at all obvious to many students (Kieran 1992; Lee and Wheeler 1987; Vergnaud 1987). To help students make the connection between algebra and arithmetic, many researchers suggest that students should be exposed to multiple methods of representing problems, including pictures, models, tables, and graphs (Dufour-Janvier, Bednarz, and Belanger 1987; Vergnaud 1987). The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989, 129–31) advocates that all students can and should learn algebra; however, some students will be capable of more abstraction than others. Encouraging multiple representations of problems is a way of accommodating students working at all levels of abstraction.
Variables and Limits
Authors: Jackson Dunham
Published: November 2020

The serious student of pure mathematics often has occasion for the uncomfortable belief that if his science is regarded with any interest at all, it is because it has ascribed to it certain qualities of doubtful merit which it does not possess.
Returns to the Questioner
Authors: Grove C. C.
Published: November 2020

There still are those who think the professor has an easy time and proceed to help make him earn his “large” salary by asking him questions, with or without a stamp for reply. It seems to be expected that he is a cistern if not a spring, but they forget at what expense a good cistern is built and a gushing spring forced.
Activities: Generating Fractals through Self-Replication
Authors: Reinstein David , Sally Paul , Camp Dane R.
Published: November 2020

Fractal geometry is often called “the geometry of nature.” To develop a feel for how fractal geometry describes natural phenomena, students can explore the process of iterating deterministic alg01ithms, which are algorithms for which the results of each step are predictable. Mathematically. fractals are the limiting result of an infinite number of selfreplications. They are often studied by using the speedy processing capability and high-resolution screen of the computer to show intermediate stages in the development of fractal images. In recent years, much interesting curriculum has been developed on this topic (see the bibliography). This activity uses a programmable graphing calculator to introduce this topic and to complement these cwTicula.
Building the Concept of Function from Students' Everyday Activities
Authors: Dauidenko Susana
Published: November 2020

In everyday life, people analyze information using algebraic thinking, often being unaware of doing so. Teachers can play an important role in helping students become aware of their own thought processes. Students should have the opportunity to bring their experiences into the mathematics class, reflect on their own thinking, and deepen their understanding of real problems.
Algebraic Thinking: A Theme for Professional Development
Authors: Ruopp Faye , Cuoco Al , Rasala Sue M. , Grace Kelemanik M.
Published: November 2020

Education Development Center (EDC), Incorporated, with support from the National Science Foundation (ESI-9253322), created a professionaldevelopment program for mathematics teachers, Teachers Time and Transformations (TTT), with algebraic thinking as its content focus.
One Good Problem Leads to Another and Another and …
Authors: Gannon Gerald E. , Bonsangue Martin V. , Redfern Terrence J.
Published: November 2020

In the September 1994 issue of the Mathematics Teacher, Krulik and Rudnick, in their article “Creative Teaching Will Produce Creative Students,” described students' reactions to the following problem:
Solutions to Calendar
Authors: McLoughlin John Grant
Published: November 2020

Problems 1–19, 28, and 29 were submitted by Margaret J. Kenney and Stanley J. Bezuszka. S.J. of the Mathematics Institute, Boston College, Chestnut Hill. MA 02167-3809. Problems 20–22 appear in The Wohascum County Problem Book by George T. Gilbert. Mark I. Krusemeyer, and Loren C. Larson Washington, D. C.: Mathematics) Association of America, 1993). Problems 23–27 were prepared by Robert Gerver and hts mathematics-research students, Sharon Poczter, Parisa Golestaneh, Nancy Friedlander, A1exis Soterakis, Nicole Miritello, Laura Henning, and Dalita Balassanian of North Shore High School, Glen Head, NY 11545.
Connecting Writing to the Mathematics Curriculum
Authors: Pugalee Dalid K.
Published: November 2020

Writing is an important aspect of the educational process in most disciplines. The power of writing as an instrument m the mathematics curnculum was realized during the 1980s as a part of the writing-across-the-curriculum movement. This interest in writing as an important learning tool is regaining popularity as reform efforts in mathematics education call for experiences “that encourage and enable students to value mathematics, gain confidence in their own mathematical ability, become mathematical problem solvers, communicate mathematically, and reason mathematically” (NCTM 1989, 123).
Applications: Approaching the Speed of Light with Class
Authors: Dance Rosalie A. , Sandefur James T.
Published: November 2020

Many of us use applications in the mathematics classroom in an effort to motivate our students. The applications also help our students understand the mathematical concepts by giving them physical objects to connect with the numbers and symbols. One problem we have had is that many applications are somewhat concocted, and the mathematics involved gives little additional insight into the situation. One application with which we have had success involves computing the speed of light in water using only some basic algebraic techniques and simple equipment that most of us have in our classrooms. In the process, students learn how mathematics can be used to make inductive discoveries of physical phenomena that would be difficult to compute directly.
Calculus Reform and Graphing Calculators: A University View
Authors: Stick Marvin E.
Published: November 2020

During the 1994–1995 academic year, I taught one section of Calculus I and II using a graphing calculator while teaching the remaining sections in the traditional way. Students were randomly assigned to the experimental sections. Evaluations of the experimental sections were conducted to determine whether the use of the graphing calculator should be considered for adoption beyond the experimental sections. The evaluations included observations of in-class student behavior, an analysis of answers to a short questionnaire administered to students during the last class before the final examination, and traditional class grades as a measure of performance. As table 1 indicates, the retention and performance for the experimental sections exceeded those of the other, traditionally taught sections.
Technology Tips: Visualizing Least-Square Lines of Best Fit
Authors: Embse Charles Yonder
Published: November 2020

Statistics and data-analysis topics are becoming an integral part of nearly every secondary mathematics curriculum. Instead of being relegated to the last, unreachable chapter, these topics are now covered early—sometimes first—in the curriculum and are used throughout to complement the study of many mathematical concepts and principles and to form the basis for a vast supply of realistic problems. Other reasons behind this signi ficant change in the focus of the secondary curriculum include the recommendations in the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) for increased attention to these topics for grades K-12 and, maybe the most important reason of all, the availability of powerful technological tools that place these Computationally intense topics within the reach of all students.
Random Variables: Simulations and Surprising Connections
Authors: Quinn Robert J. , Tomlinson Stephen
Published: November 2020

Integrating probability and statistics within the curriculum provides a number of interesting and elegant connections that help students develop an appreciation for the inherent beauty of mathematics. This lesson on random variables incorporates class discussion and experimental activities in the practical and theoretical exploration of one such connection.
In Search of Perfect Triangles
Authors: Bonsangue Martin Vern , Gannon Gerald E. , Buchman Ed , Gross Nathan
Published: November 2020

Working on the perfect–triangle problem was a real mountaintop experience in mathematics. Our problem–solving path started with basic geometry, followed by a little algebra, then more geometry and more algebra. Our tour was enhanced by using a computer not to solve the problem but to explore its solution.
Soundoff!: A New Breed of Calculators: Do They Change the Way We Teach
Authors: Podlesni James
Published: November 2020

A new generation of graphing calculators—for example, the TI-89 and the Casio 9970—that use computer algebra systems (CAS) are now available. Since they manipulate symbols, one could argue that they represent as big a change as the step from scientific to graphing calculators. A student “asks” the calculator to factor x2 + 5x + 6, and the calculator prints (X + 2)(X + 3). On the Advanced Placement calculus examination, existing graphing calculators allow the student to find the derivative of y = cos x at a specific x-value. In addition, the new models can “tell” the student that the derivative of cos x is −sin x. They can also provide units for numeric answers. They are apparently much more user-friendly than the existing HP-48, thereby assuring widespread use. The TI-89 is essentially a TI-92 in a TI-83 case, without the geometry package but with FLASH™, a feature that allows it to be upgraded electronically.
The Amazing Octacube
Authors: Naylor Michael
Published: November 2020

This octacube makes an excellent classroom model that will fascinate and challenge your students to explore further many of the intriguing properties and relationships of the Platonic polyhedra.
Sharing Teaching Ideas: We Have Liftoff! Introducing the Logarithmic Function
Authors: Hurwitz Marsha
Published: November 2020

The Curriculum and Evaluation of Standards for School Mathematics (NCTM 1989) targeted the concept of function as one that should receive increased attention. A function can be defined as a rule that assigns to every input value exactly one output value. To make it easier for novices to recognize a function, we should select the rule, the input, and the output to be straightforward.
Build a Sierpinski Pyramid
Authors: Kelley Paul
Published: November 2020

Students from Anoka High School, located in Anoka, Minnesota, built a nineteen-foot-tall Sierpinski pyramid in the Minneapolis Convention Center in conjunction with the NCTM's 75th Annual Meeting in April 1997. This activity was part of a unit on fractal geometry.
Discovering Optimum Networks in Triangles
Authors: Iovinelli Robert
Published: November 2020

This classroom activity is an example of how students can use the connections among different disciplines of mathematics to investigate a branch of mathematics that is not usually encountered in high school courses—graph theory. The first graphs in this activity are triangles, and the sides of the triangles form a planar network. Other line segments can connect the three vertices of the triangle, forming other networks—the lesson's main objective is to find the optimal “connecting network” for a given set of three vertices—the network made up of line segments with the shortest possible total length.
Making the Black Box Transparent
Authors: Lesser Lawrence M.
Published: November 2020

Although introducing technology into our mathematics curricula allows us to tackle problems of size and complexity as never before, we face a danger of introducing tools to students before they have a sufficient understanding of how mathematics content within their reach can be used to shed light on the algorithms within the tools or on the use of the tools themselves. Fortunately, we can view mathematical theory and technology not as opponents but rather as partners that make the whole of mathematical understanding richer than the sum of its parts. Indeed, bringing technology into our classrooms can encourage new questions that technology-free mathematics must answer. This article focuses mainly on a common example in technology-rich mathematics curricula, namely, the line of best fit, followed by a discussion of two additional examples—interpolating polynomials and complete graphs. In each case, connections between theory and technology do not appear to be as widely known and used as they could be.
Geometry's Giant Leap
Authors: Brown Alan R.
Published: November 2020

After many years during which graphing calculators were used for algebra and other advanced mathematics subjects, dynamic software for geometry is finally available. Such software programs as The Geometer's Sketchpad and Geometric Supposer have existed for several years; however, they require expensive hardware and dedicated space and have limited transportability. Technology for most geometry classes has traditionally consisted of a compass, a straightedge, pencils, and, more recently, a scientific calculator. However, the TI-92 graphing calculator is the first of a new breed of handheld calculators with dynamic software.
Projects: Hawaii Algebra Learning Project
Authors: Kasten Peggy
Published: November 2020

The Hawaii Algebra Learning Project of the Curriculum Research and Development Group, University of Hawaii, has both a curriculum component and a professional development component. The project was made possible through funding from the National Science Foundation, the U.S. Department of Education, the McInerny Foundation, and the University of Hawaii. The original goal of the project was to develop curriculum materials that make algebra accessible to students of all ability levels. This materials component has been identified as a promising program in mathematics education by the Laboratory Network Program of the U.S. Department of Education's Office of Educational Research and Improvement.
Anatomy of an Assessment
Authors: Long Vena M.
Published: November 2020

The Assessment Standards for School Mathematics (NCTM 1995) discusses a shift in mathematical content from just mastering isolated concepts and procedures to exploring a rich variety of mathematical topics and problem situations that teach and use concepts and procedures to solve problems. The Standards document also calls for student assessment that is aligned with, and integral to, instruction. Constructing valid assessments has always been difficult, and the new demands make it even more of a challenge. What does such assessment look like, and how does it work?
Analytic Methods in Investigative Geometry
Authors: Dobbs David E.
Published: November 2020

In the Mathematics Teacher's 1998 focus issue on the concept of proof, Stanley P. Izen (1998) argues that computer-based discovery activities in a geometry course should be followed by rigorous proofs of the conjectures developed during those activities.
Media Clips: January 2002
Authors: Collier C. Patrick
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 full-blown projects for optional individual pursuit. Each lesson, though, should help convince our students that mathematics is useful, current, and newsworthy. Please send submissions to –Media Clips,– Mathematics Teacher, 1906 Association Drive, Reston, VA 20191-9988.
Hermione Granger's Solution
Authors: Howe Roger
Published: November 2020

N THE MIDDLE OF THE CLIMACTIC EPISODE of J. K. Rowling's phenomenally popular children's novel Harry Potter and the Sorcerer's Stone, Harry Potter and Hermione Granger find themselves in a room whose two exits are blocked by raging flames. On a table in the room stand seven bottles, along with a piece of paper inscribed with the following verses.
Discuss with Your Colleagues: Exploring the Four-Points-on-a-Circle Theorems with Interactive Geometry Software
Authors: Jiang Zhonghong , Pagnucco Lyle
Published: November 2020

Some theorems and converses of theorems are not treated adequately in many of our school textbooks and resources; this lack of attention leaves a sense of incompleteness when considering the rigor and sense making of mathematics.
Mind Mapping as a Tool in Mathematics Education
Authors: Brinkmann Astrid
Published: November 2020

Mind mapping is a special technique for taking notes. Ideas and concepts connected with a topic are displayed as a graphical pattern—in many instances, even as in an artistic image. The mind-mapping technique takes into account the special way that the human brain works and thus facilitates the organization of information and increases productivity and memory retention.
Early Secondary Mathematics: Teaching Sums of Angle Measures: A Kinesthetic Approach
Authors: Touval Ayana , Westreich Galeet
Published: November 2020

The wide acceptance of Howard Gardner's multiple intelligence theory (Gardner 1983) and discussions of bodily kinesthetic intelligence have prompted much research and led to the adoption of a variety of new teaching approaches. The kinematics teaching strategy (KTS) is a new approach for stimulating kinesthetic intelligence. It was first developed to respond to the needs of children who encounter difficulties when taught only with auditory and visual methods. As the following lesson illustrates, KTS can be used successfully in a broader context.
Media Clips: April 2003
Authors: Harlos Carol Ann
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 full-blown projects for optional individual pursuit. Each lesson, though, should help convince our students that mathematics is useful, current, and newsworthy. Please send submissions to “Media Clips,” Mathematics Teacher, 1906 Association Drive, Reston, VA 20191–1502.
A Direct Approach to Computing the Sine or Cosine of the Sum of Two Angles
Authors: Gutmann Timothy
Published: November 2020

When I teach precalculus, I struggle to decide how much to emphasize different trigonometric identities. Knowing them can help students in subsequent courses, and working with them exposes students to richer applications of algebra than solving one- or two-variable polynomial equations. Yet I often find that the derivations of trigonometric identities are out of place in my classroom, where the focus is on understanding functions as tools for modeling realworld phenomena. There, sine and cosine values are tied directly to measurements of segments within the unit circle.
Sharing Teaching Ideas: How Can We Find the Coefficients From the Graph y = ax2 + bx + c Using Geometric Tools Only?
Authors: Satianov Pavel
Published: November 2020

Nowadays, with the help of a computer or graphing calculator, we can easily obtain an accurate graph of a function if we are given the formula that represents it. However, suppose that we are given the graph of a function and asked to determine the parameters of the formula that generated it. We would like to share one method of finding the coefficients of the quadratic function. The method may differ from methods that readers have previously seen (Metz 1994). Our approach describes a property of the parabola that is connected with the main coefficient a in the standard representation y = ax2 + bx + c.
How to Solve Any Triangle: First, Forget the Law of Sines and the Law of Cosines
Authors: McMullin Lin
Published: November 2020

Computer algebra systems (CASs) have made, and will continue to make, changes in what is taught in mathematics courses at all levels. Just as scientific calculators made teaching computations with logarithms obsolete, CASs will render obsolete many other topics in the curriculum. Some will be formulas and procedures for computations that are no longer necessary to solve the problems for which they were originally developed. The mathematical problems will remain; we will just use more efficient ways to solve them. A good example is Newton's method for approximating the zeros of an expression. Using Newton's method to find roots by hand has been rendered obsolete by graphing calculators that can either find the roots exactly or do the approximations faster and more accurately.
Projects: Upper Midwest Leaders-in-Math Project: Preparing Leaders for Education and Professional Development in the Effort to Renew School Mathematics
Authors: Foletta Gina , Becker Joyce
Published: November 2020

Implementing comprehensive NCTM Standards-based reform can be both challenging and fulfilling. Seeing documented improvements in the teaching and learning of mathematics for all students, as shown by a growing body of research, is fulfilling. For example, researchers for the elementary school Alternatives for Rebuilding Curricula (ARC) Tri-State Student Achievement Study state, “The principal finding of the study is that students in the NSF-funded reform curricula consistently outperformed the comparison students…. This result held across all tests, all grade levels, and all strands, regardless of SES and racial/ethnic identity” (ARC 2002, p. 5). In a recent volume of collected research on NSF-funded reform curricula, a reviewer of the K—12 research states, “The studies reported in this book provide much needed evidence that the new programs work” (Kilpatrick 2003, p. 472); and a reviewer of the high school research states, “[There is] considerable evidence that the promises of reform mathematics are real and the fears of the anti-reformers unjustified” (Swafford 2003, p. 458).
Fostering Mathematical Curiosity: Highlighting the Mathematics
Authors: Knuth Eric J. , Peterson Blake E.
Published: November 2020

In “Fostering Mathematical Curiosity” (Knuth 2002), Eric Knuth discusses the idea of problem posing as a means of fostering students' mathematical curiosity. A mathematician colleague, after reading the article, commented that a significant amount of the mathematics in the discussion of the various problems and their solutions had been “left out” —and he was right. The author's primary intent in writing that article, however, was to illustrate “what it might mean to engage students in problem posing and how teachers might begin to create classroom environments that encourage, develop, and foster mathematical curiosity” (Knuth 2002, p. 126), not to discuss in detail the mathematics underlying the solutions to the problems posed.
Sharing Teaching Ideas: Linear Equations: High School Algebra 1
Authors: Foss Rita G.
Published: November 2020

Teaching linear equations is an important part of algebra 1—and the topic involves so much information that students can become overwhelmed and frustrated with the sheer volume of notes, formulas, and other “stuff” to remember. Algebra 1 students, like students in any entry-level course, need help organizing notes, discriminating important information, and finding a way to remember formulas, calculator steps, and relationships. Teachers new to teaching entry-level courses frequently assume that special education students need more concrete learning experiences or that English Speakers of Other Languages (ESOL, or limited English) students have more vocabulary needs. These assumptions are often correct, but I have discovered that most high school algebra 1 students have the very same needs. Many have not been successful in previous mathematics classes, and few know how to pick out important concepts and relate them to other ideas. To help students keep track of the information needed in studying lines, we make a flip chart that the student can keep and use throughout the year, as well as in succeeding years.
Motivating Activities That Lead to Algebra
Authors: Menon Ramakrishnan
Published: November 2020

Activities with puzzles and algebraic patterns are used to motivate students to learn algebra and assist understanding for preservice teachers and students. Included are directions and student work for the constant sum grid, Fibonacci sequence, predict your age, and target 21 games.
Patterns in Perfect Squares: An Activity for Exploring Mathematical Connections
Authors: Farmer Jeff D. , Neumann Andrew M.
Published: November 2020

A versatile mathematical problem that begins by asking students to find patterns in a list of perfect squares. The representations and explanations of those patterns can lead to connections between many mathematical concepts. The patterns are represented algebraically, geometrically, and pictorially. Patterns with multiples and arrangements are explored while developing reasoning and communication.
Drug Levels and Difference Equations
Authors: Acker Kathleen A.
Published: November 2020

Students examine details of Tylenol. Using difference equations students develop mathematical models for the amount of a drug in the body after a single dose and after multiple doses. They compare other data about the drug and create geometric growth models.
Delving Deeper: Fibonacci and Lucas Numbers
Authors: Askey Richard A.
Published: November 2020

In “Fibonacci and Related Sequences” (Askey 2004), we investigated identities involving Fibonacci and Lucas numbers, which are defined by the relations.
Math through the Mind's Eye
Authors: Fisher Sara P. , Hartmann Christopher
Published: November 2020

This paper considers recommendations from the Principles and Standards for School Mathematics (PSSM) in relation to pedagogy for the visually impaired. The authors present three examples of ways that mathematics instruction for blind learners can employ representations in ways that are consistent with PSSM. In reflecting on these examples, the authors identify lessons for all mathematics teachers. The nature of these accommodations provide a new perspective on the recommendations in the PSSM.
Paper Moon: Simulating a Total Solar Eclipse
Authors: Madden Seán P. , Downing James P. , Comstock Jocelyne M.
Published: November 2020

This article describes a classroom activity in which a solar eclipse is simulated and a mathematical model is developed to explain the data. Students use manipulative devices and graphing calculators to carry out the experiment and then compare their results to those collected in Koolymilka, Australia, during the 2002 eclipse. Includes a description of how to set up the simulation and examples of student work.
A Hands-On Approach to Teaching Composition of Functions to a Diverse Population
Authors: Boognl Mary A.
Published: November 2020

Teachers who have decided that every student will have a chance to learn find that they must design lessons that reach each classroom's diverse population. Although the lesson presented in this article appeals to students of all ethnic groups and learning styles, it specifically meets the needs of my target population, Native American students, because it provides for (1) cooperative learning groups, (2) visual learners, (3) self–paced learning, and (4) the natural emergence of language.
CREEP OF POLYCRYSTALLINE IRON UNDER SMALL TORSIONAL STRESS AND THE EFFECT OF CARBON UPON THE CREEP
Authors: KE TING-SUI  , KUNG CHING-PING  ,   中国科学院金属研究所
Published: November 2020

Creep under very small torsional stress has been investigated for 99.95% polycrystalline iron in the temperature range of 350°-525℃. The general features of the creep curves obtained are essentially the same as those previously reported for 99.991% aluminum. The creep is shown to be consisted of two parts, one occurring at lower temperatures or shorter times and the other at higher temperatures or longer times. The first part of the creep is limited in extent and is associated with the viscous behavior of grain boundaries. The activation energy for this grain-boundary creep is 78,000 ± 4,000 calories per mole according to the present creep and stress relaxation measurements. This value is close to the activation energy for self-diffusion of iron, a correlation which indicates that, in so far as to the mechanism of atomic migration is concerned, grain boundaries may not be very different from the interior of the grains.Both parts of the creep are noticably lessened by the addition of carbon in the iron specimen. A striking feature is that the viscous slip along grain boundaries appears to be hindered by carbon at a concentration as low as 0.0004%. These observations provide a guiding clue as to the problem of controlling high-temperature creep of metals and render further experimental support on the hole model previously suggested for the grain boundaries.With regard to the second part of the creep, it is tentatively suggested that its mechanism may be connected with atomic rearrangements of holes inside the grains.
ON THE PROBLEM OF AXIAL-SYMMETRIC PLASTIC DEFORMATION
Authors: LIN HUNG-SUN  ,   中国科学院数学研究所
Published: November 2020

In the usual treatment of plane deformation problems in plasticity, the assumption of static determinacy is frequently made, with the advantage of obtaining stress distributions without reference to stress-strain relations. In the present paper, we consider the problem of axial-symmetric deformation, where this advantage is no longer existing. The material is assumed to be incompressible and entirely in a plastic state. The equations of equilibrium are
SOME PRACTICAL PROBLEMS ARISING FROM THE ADOPTION OF THE ABSOLUTE SCALE OF TEMPERATURE AS STANDARD
Authors: WANG JWU-SHI  ,   北京大学物理系
Published: November 2020

Some practical problems arising from the adoption of the absolute scale ol temperature as standard in place of the Celsius scale are discussed. The absolute scale is fixed by assigning a numerical value (say T3=273.1700) to the triple point of water. The problems 'considered include the correction of gas thermometer readings, the specification of: the platinum resistance thermometer and of the platinum thermocouple. It is pointed out that, when the absolute scale has been chosen as standard, the determination of the absolute temperature of the ice point and the correction of gas thermometers become one and the same task. It is further pointed out, that the accuracy in the experimental determination of the absolute temperature will be greatly increased.
A NEW METHOD OF CALCULATION FOR THE STRUCTURE OF MOLECULES
Authors: PENC HUAN-WU  ,   中国科学院物理研究所
Published: November 2020

A new method of calculation of the self-consistent field type is outlined for studying the electronic structure of molecules. In this method the fundamental wave functions refer to individual bonds of the molecule and not to individual electrons as in the existing methods of atomic orbitals and molecular orbitals.
A MAGNETIC INTERNAL FRICTION PEAK IN NICKEL UNDER ALTERNATING MAGNETIC FIELD
Authors: KAREL MISIEK  ,   中国科学院金属研究所
Published: November 2020

The coupling between elastic and magnetic phenomena in ferro-magnetic materials often gives rise to internal friction. In previous studies on magnetoelastic internal frictions, the applied magnetic field was mostly static. In the present experiment, nickel specimen was made to vibrate in torsion under an alternating magnetic field, and a pronounced internal friction peak was observed when internal friction was plotted as a function of the alternating magnetic field strength. This peak did not appear under similar conditions when using a static magnetic field. The position of the peak was shifted toward the side of a higher magnetic field when the nickel specimen was either cold-worked or contained carbon; and it was shifted toward the side of a lower magnetic field when the specimen was previously annealed at an elevated temperature. The height of the peak was considerably raised with an increase of vibrating frequency.A qualitative explanation of this peak is given on the basis of reversible and irreversible processes occuring at different stages of the magnetization curve of nickel. Factors affecting this peak arc discussed.
DIFFUSION AND PRECIPITATION OF CARBON AND NITROGEN FROM SOLID SOLUTION IN IRON AND STEEL
Authors: KE TING-SUI  , YUNG PAO-TSUI  , WANG YEH-NING  ,   (1)中国科学院金属研究所; (2)中国科学院金属研究所,南京大学物理系讲师中国科学院金属研究所
Published: November 2020

It is well-known that blue brittleness and temper brittleness of steel are related to diffusion and precipitation in steel. An attempt was made in the present research to investigate this problem by means of internal friction measurements. Experiments show that the diffusion of carbon in α-iron is not affected by carbon concentration, alloying elements and the stage of precipitation in iron. The diffusion of nitrogen in α-iron is, however, hindered by the addition of alloying elements, and is enhanced during the initial stage of precipitation.The internal friction peak (around 250℃ for a frequency of vibration of 1 cycle per second) associated with the precipitation of carbon or nitrogen in a cold worked a-iron specimen previously observed was further studied. It is found that in specimens of sufficiently high carbon or nitrogen content, the internal stress resulting from the quenching of the specimens from high temperatures also gives rise to this internal friction peak. Further experiments show; (1) this internal friction peak is associated with the phenomenon of strain aging or quench aging, and it occurs under conditions corresponding to the appearance of blue brittleness in steel; (2) the carbon or nitrogen associated with this internal friction peak is in a dispersed atomic form. This correlation suggests that the blue brittleness of steel is a process of strain aging or quench aging and it is directly associated with the separation of carbon or nitrogen atoms from solid solution into stress regions or defects in steel.A tentative suggestion concerning the mechanism of temper brittleness in steel has been made that temper brittleness is associated with the process of precipitation of nitrides instead of carbides in steel.
THE FRICTION-FIELD OF LEAST RESISTANCE
Authors: LIU SHU-I  ,   中国科学院冶金陶瓷研究所
Published: November 2020

The contact surface in plastic deformation is considered as a field, whose field-line is given by the "generalized rule of gradient" based upon the condition of least resistance applied both to effective and ineffective direction of "relative" slip on solid and liquid surfaces, as shown by equation (1),which is the formulation of the sta tement that the direction of field-lines is the direction of (most rapidly) decreasing frictional resistance [equation (2) ]. The main results are:(1 )The rule of field lines is given by equation (l)and(2). Accordingly, for dry friction, the"resistant" field is divergent; for plastic lubrication, the "lubricating" field is convergent (Fig. 1). There may be combination of both types, as shown in Fig.l(c) and Fig. 4 for the case of plate-rolling. (2)The two field functions, the unit friction (τ) and the unit pressure (p) are governed by equations (3), (4), (8), (9), (11), given in this work, by equations (5), (6), (7), (15) of classical theory and by equations (6), (12), (14), (16)and Fig. 6, all based upon classical concept, but first used in our work. Fig. 6 deals with range of variation of what is termed the "point of friction", that is, (τ, p) point in the stress plane. Equation (12) reads: The point of friction (Q) must be on the segments of the curve τ=Fp, merged in the closed field limited by the three stress circles where F is the "friction-function" defined by equation (6). In these equations, S is are-length of field lines. The other designations are self-evident. (3) A region of sticking in plate-rolling is determined by equation (15) as shown in Fig.6. It is pointed out that the value of the coefficient of friction can be measured by two separate experiments on pressure distribution and on friction-lines.The friction-field of least resistance provides a new fundation for plasticity under pressure.
INITIAL ENERGY DISTRIBUTION AND ANGULAR DISTRIBUTION OF PHOTOELECTRONS
Authors: WU CHUAN-TEH  ,   北京大学
Published: November 2020

In the present paper, the Nottingham's Empirical Formula of the transmission coefficient is used to develop the DuBridge's theory of initial energy distribution of photoelectrons from metal. Owing to the detailed considerations of refraction effect for an electron passing through the potential barry of the surface, we discuss not only the energy distribution and the normal energy distribution of photoelectrons, but also the angular distribution. All discussions are based on the following assumptions:(1) The surface is ideal;(2) The energy distribution of electrons in the metal is according to the Fermi-Dirac distribution;(3) The intensity of incident monochromatic light is constant and the incident light is unpolarized.The obtained formula of initial energy distribution of photoelectrons is very simple, but it is more satisfactory than the other theories shown in this paper. The curve of angular distribution of photoelectrons is like an egg with its large end outward. If there is no refraction effect, the angular distribution satisfies Lambert's law. The spectral distribution, the theoretical formula of volt-ampere characteristic under the retarding field of a spherical condenser with a small photocathode as one electrode at centre, the normal energy distribution, and the theoretical volt-ampere characteristic of ideal parallel plate condenser are also discussed in this paper.
THE INTERFERENCE PHENOMENA OF LIGHT AT VERY LOW INTENSITY
Authors: L. JANOSSY  , Zs. NARAY  ,   匈牙利科学院中央物理研究所
Published: November 2020

An interference pattern is obtained with a Michelson interferometer; the intensity distribution in the pattern is determined by counting the rate of photons by means of a photomultiplier. It is shown that the pattern obtained for very low intensities of light does not differ outside the margin of experimental error from the pattern obtained for normal intensities. At low intensities about 106 photons enter the interferometer per second thus for these low intensities in average at any time much less than one photon is contained inside the interferometer.
Laboratory study of non-ideal effects in magnetically collimated astrophysical outflows
Authors: Tao Tao ,   Department of Engineering and Applied Physics, University of science and technology of China, Hefei 230000, China
Published: November 2020

Central outflow’s collimation by magnetic field is an important theoretical mechanism for explaining the astrophysical objects’ morphology formation, and its credibility has been tested in many laser plasma experiments in a dimensionless manner. This article introduces integrated simulation and experiment work based on the present laboratory magnetically collimated jet framework, to explore how non-ideal terms’ strength including radiative cooling and magnetic diffusion from different targets can affect the outflow shape. The interaction between outflow from a target with low atomic number and external field satisfies the ideal magneto-hydrodynamic conditions, and the outflow shape results in diamagnetic cavity and jet; on the other hand, a heavy element target brings strong magnetic diffusion that destroys the collimation structure, together with the stagnation of outflow introduced by radiative cooling, and outflow shape results in weakly collimated hemisphere near the target and a detached magnetized density clump. The detailed dimensionless analysis shows that the large-scale dissipation of jets in young stellar objects can possibly be an analog of the laboratory jet’s magnetic diffusion breakup, also similar structures like the loosely collimated lobes and bright ansaes in planetary nebula can be observed in highly diffusive laboratory outflows. This article shows for the first time that a series of non-relativistic astronomical outflows’ dynamic behaviors can be explained by the non-ideal magneto-hydrodynamic evolution of laboratory plasmas.