bluuebunny/crossref_metadata_2025 huggingface dataset:
Generalized Fibonacci Sequences
Authors: Bradley Sean
Published: November 2020

Everyone loves the Fibonacci sequence. It is easy to describe, yet it gives rise to a vast amount of substantial mathematics. Physical applications and connections with various branches of mathematics abound. What could be better, unless someone told us that the Fibonacci sequence is but one member of an infinite family of sequences that we could be discussing? The generalization that follows has great potential for student and teacher exploration, as well as discovery, wonder, and amusement.
Word Histories: Melding Mathematics and Meanings
Authors: Rubenstein Rheta N. , Schwartz Randy K.
Published: November 2020

Integrating the history of mathematics into instruction sometimes seems to require taking advanced courses or studying lengthy texts. However, much of mathematics history is reflected in the very words that we use every day. The etymologies, or origins, of mathematics words make a rich resource for deepening students' understanding and appreciation of mathematics, history, and language. They open a window onto the lively history of our science and its connections with other subjects. In this article, we share some of these etymologies, as well as ideas for incorporating them into instruction.
The Evolutionary Character of Mathematics
Authors: Davitt Richard M.
Published: November 2020

This article advocates Grabiner's UDED paradigm [use-discover-explore-define] as a tool for teachers' own acquisition of authentic historical accounts of the evolution of mathematical topics and as a pedagogical strategem for their students as well.
Mathematicians Are Human Too
Authors: Lightner James E.
Published: November 2020

WE OFTEN STUDY THE PUBLISHED WORKS of the great mathematicians, accept them, and use them gratefully as we solve our problems and delve into the abstractions of our chosen field of mathematics. But rarely do we realize that all these developments were the products of human minds. Mathematicians, indeed, are quite human, and they are not always serious! Their lives are full of interesting twists, turns, and quirks that make them all the more human.
From the Top of the Mountain
Authors: Smith Donald W.
Published: November 2020

This article describes how John Napier published the first view of logarithms and how this concept has been transformed—beginning during his own lifetime—into something quite different. Seeing Napier's work through the lens of our present interpretation of logarithms is tempting, but the actual story is every bit as impressive. The quotation from Moulton offers a useful perspective about the path that Napier took.
Felix Klein and the NCTM's Standards: A Mathematician Considers Mathematics Education
Authors: McComas Kim Krusen
Published: November 2020

The year 1999 marked the 10th anniversary of the NCTM's Curriculum and Evaluation Standards for School Mathematics. It also marked the 150th anniversary of the birth of German mathematician Felix Klein, who lived from 1849 to 1925. Although the relation between these two anniversaries may not be obvious, the connection is that Klein, were he still alive today, would probably support the NCTM's Standards. As the year 2000 brings us NCTM's Principles and Standards for School Mathematics, let us look back to the year 1900 and find Felix Klein at the forefront of a movement to reform mathematics education from rote learning to more meaningful mathematical learning.
Primitive Living Structures: Tents and Tipis
Authors: Green Roger A. , Snyder Laura A.
Published: November 2020

Most children and even a few adults have stretched cord between two trees, thrown a blanket across the top, and staked down the sides to fashion a makeshift tent. The results are rarely pretty, but children are normally undeterred by their own imperfect workmanship.
Activities for Students: The Volume of a Pyramid: Low-Tech and High-Tech Approaches
Authors: Albrecht Masha
Published: November 2020

This lesson came about spontaneously during a geometry unit on volume. I had used the lesson shown here in activity sheet 1, in which students use cubic blocks to rediscover the formulas for volumes of right prisms, that is, V = Bh and V = lwh. This lesson was a simple review for my tenth-grade class, and they completed it easily before the end of the period. With the wooden cubes still on their desks, most of them used the remaining time to build towers and other objects. I noticed that many students piled the cubes into bumpy pyramidal shapes. Because the next day's lesson involved studying the volume of pyramids, I wondered whether these bumpy shapes could be useful for discovering the volume of a real pyramid with smooth sides. Students could compare the volumes of these “pyramids of cubes” with the volumes of corresponding right prisms and perhaps discover the ratio 1/3 to obtain the formula for the volume of a pyramid, V = (1/3)Bh. As it turns out, the ratio of 1/3 does not become evident right away. To my students' delight, we found that using a spreadsheet is an excellent way to investigate this problem. My geometry classes had not used spreadsheets before, and the students enjoyed the experience of using the efficiency of technology to compare hundreds— and even thousands—of shapes with ease.
Dinosaurs, Dinosaur Eggs, and Probability
Authors: Teppo Anne R. , Hodgson Ted
Published: November 2020

In their article “What Every High School Graduate Should Know about Statistics,” Scheaffer, Watkins, and Landwehr (1998) contend that one cannot understand statistics without understanding probability. As a consequence, the authors outline several recommendations regarding teaching probability in the secondary school.
Under Construction: On Becoming a Constructivist in View of the Standards
Authors: Ward Cherry D.
Published: November 2020

Since the beginning of time, trends in the philosophy and methods of pedagogy have come and gone, with only minor variations from traditional instructional methods. These trends have included progressive movements, essentialist movements, teacher-centered instruction as opposed to studentcentered instruction, drill and practice, projectbased instruction, discovery learning, and many other movements (Cuban 1993). In his studies, however, Cuban found that teachers have basically taught the same way, with little variation, despite these instructional trends.
Points on the Path to Probability
Authors: Kiernan James F.
Published: November 2020

Many textbooks on probability open with the story of how the Chevalier de Méré approached Pascal with the problem of points. Very few textbooks explain the nature of this problem, which requires determining the division of stakes when a game is ended prematurely. Still fewer books state that the problem was several centuries old when Pascal solved it. Many students are under the misapprehension that all mathematical knowledge is the result of some single brilliant idea rather than a cumulative process of successes and failures. A historical examination of the various attempts to solve this problem can give students insight into the nature of mathematical discovery.
Just Thinkin' of the Rain
Authors: Emenaker Charles E.
Published: November 2020

A group project that involves geometry and falls into an area of mathematics known as operations research.
Activities for Students: Seeing How Money Grows
Authors: Metz James
Published: November 2020

Compound interest is very much a part of the life of every student. Albert Einstein once called the compound interest formula the most important formula in mathematics. Unfortunately, many students never see the formula unless they take enough mathematics to encounter it in their study of exponential functions. Even then, they may experience it only algebraically, not graphically. Its predecessor, the simple interest formula, is usually studied in isolation, often in a “consumer math” application as an exercise in arithmetic or perhaps in algebra. Few students realize that the simple interest formula can be modeled with the graph of a linear equation when the principal and interest rate are held constant. In a similar way, the compound interest formula can be modeled graphically as an exponential curve. Any student who can make tables and plot points can graph the future value of an account using simple and compound interest and can thereby come to appreciate the meaning of the variables in each formula and how those variables contribute to the growth of an account.
When Memory Fails: Putting Limitations to Good Use
Authors: Cuoco Al , Manes Michelle
Published: November 2020

How you can frequently use the memory limitations of a graphing calculator to teach important conceptual ideas.
Triangles à la Fettuccine: A Hands–on Approach to Triangle–Congruence Theorems
Authors: Eggleton Patrick J.
Published: November 2020

An active approach to learning the triangle–congruence theorems.
Activites for Students: Dynamic Diagrams
Authors: Bremigan Elizabeth George
Published: November 2020

Reasoning with visual representations is an important component in solving many mathematical problems and in understanding many mathematical concepts and procedures. Students at all levels of mathematics frequently encounter visual representations—for example, diagrams, figures, and graphs—in discussions of mathematical ideas, in mathematics textbooks, and on tests. Teachers often use visual representations in the classroom when they present a mathematical problem, explain a problem's solution, or illustrate a mathematical concept. Although they frequently encounter and use visual representations in the mathematics classroom, neither teachers nor students may explicitly recognize the power of reasoning with visual representations or the potential for misconceptions that can arise from their use.
Sound Off!: Experimental Mathematics: Self-Contradiction or Lifeblood?
Authors: Epstein David , Levy Silvio
Published: November 2020

Mathematics is supposed to be the result of pure thought and rigorous proof.
Real-World Problems as Contexts for Proof
Authors: Hodgson Ted , Riley Kate J.
Published: November 2020

When one thinks of proof, one typically thinks of the abstract, structural aspects of mathematics. Proof is used, in conjunction with logic, to establish the truth of mathematical statements and to connect these statements with others in mathematics. At the secondary level, students prove statements in geometry, algebra, or calculus. These “truths” can then be used to solve real-world problems. However, proof and real-world problem solving are typically considered to be separate and distinct endeavors.
Media Clips: December 2001
Authors: Patrick Collier C.
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.
Projects: UTeach: Secondary Science and Mathematics Teacher Preparation Program
Authors: Foletta Gina
Published: November 2020

The UTeach program, funded by the National Science Foundation, constitutes a joint effort of the College of Natural Sciences, the College of Education, and the Austin Independent School District to recruit, prepare, and support a new generation of mathematics and science teachers for Texas and for the United States.
Sharing Teaching Ideas: Letting the Cat out of the Bag … To Make Room for a Triangle!
Authors: Reynolds Marie J.
Published: November 2020

If Dr. Seuss can touch millions of hearts with his Cat in the Hat, then geometry teachers should be able to tickle the brains of high school students with this “triangle in the bag.” On the first day of our triangle-congruence chapter, I challenge my geometry students to duplicate the triangular shape that I have hidden in a brown paper bag. Their assignment is to silently write questions about the unseen triangle, raise their hands, and seek written answers from me. This investigative approach to congruence leads to amazing results in a short period of time.
Technology Tips: January 2002
Authors: Starnes Daren , Gosse Paul
Published: November 2020

We are especially pleased to offer a technology tip for Hewlett-Packard users this month. Colin Croft, a colleague from Australia, shows some potential pitfalls in using the HP—or any other calculator, for that matter—to evaluate limits. Next, coeditor Paul Gosse gives an inside look at Graph Databases (GDBs). If you have not used a GDB before, you might try to do so after you read his suggestions. In our third piece, Bob Ruzich illustrates some nice calculus extensions of the classic “ball bounce with a motion detector” activity. Appropriately enough, our featured Web site this month is Colin Croft's own site for HP calculator users.
Ratio Analysis: Where Investments Meet Mathematics
Authors: Barton Susan D. , Woodbury Denise
Published: November 2020

A valid argument can be made that concepts in mathematics should be studied for their own intrinsic value. Yet almost every high school mathematics teacher has heard the plaintive wail, “When will I ever use this?” Many students need to ground their understanding of mathematical concepts in a realworld setting. NCTM's Principles and Standards for School Mathematics also confirms that developing connections among mathematics, other subjects, and the world outside the classroom is important in giving the students “greater mathematical power” (2000, p. 354).
Move That Sofa!
Authors: Vennebush G. Patrick
Published: November 2020

What is the largest sofa that can be moved through a corridor of unit width that contains a right-angled turn? One might naively think that the solution is easy to discover—yet this problem remains unsolved.
The Dangers of Mathematical Modeling
Authors: Köhler Angela D. A.
Published: November 2020

When setting long-term goals, mathematics teachers face a constant dilemma. Most of us realize that our students will need to be mathematically literate in their future jobs, be able to see the real world through mathematical eyes, and be ready to handle the huge quantities of numbers that will be presented to them in their company's reports and in the news. During most of the school year, however, we give our students problems that are already written in mathematical language. Even the socalled real-life applications often consist of just an equation from physics, medicine, or economics that students are expected to analyze algebraically or graphically. They do not learn how the equation was originally derived, and they can often solve such problems without giving any thought to the application. To truly connect their mathematical skills with the outside world, we need to confront them with problems that have not yet been translated into the language of mathematics. We should just say “Now what?” and give them the necessary time to try out several mathematical models for the given situation.
Technology Tips: February 2002
Authors: Starnes Daren
Published: November 2020

In last month's column, Bob Ruzich described how to analyze velocity data collected from a CBL/CBR ball-bounce experiment using the ΔList feature on the TI-83 graphing calculator. This month, Ruzich returns to share strategies for verifying the law of exponential change using a CBL or CBL 2 voltage experiment. Next, coeditor Paul Gosse gives you a fresh look at teaching linear programming with the TI-83 and spreadsheet software. Be sure to look at some of the fine differences that Paul discusses between “free-cursoring” and tracing. Our “Surfing note” takes you to Japan for a look at some dynamic, interactive Java applets.
Socratic Seminars for Mathematics
Authors: Koellner-Clark Karen , Stallings L. Lynn , Hoover Sue A.
Published: November 2020

Stephanie and Quavis gestured animatedly as they stood in front of the graphs, shown in figure 1, that had been drawn on the board. Tara and Laura were speaking from their desks, and the rest of their second-year algebra class watched intently. We will subsequently share more of their discussion as we discuss an example of a Socratic seminar, but we first notice how these students struggled to “organize and consolidate their mathematical thinking through communication” and to “communicate their mathematical thinking coherently and clearly to peers, teachers, and others” (NCTM 2000, p. 348). The students were arguing vigorously and seemed disappointed when the bell rang to end the seminar.
Statistical Sampling and Data Collection Activities
Authors: Gelman Andrew , Nolan Deborah
Published: November 2020

One way that students learn about sampling is by actually collecting data. Collecting data gives them a feel for the practical struggles and small decisions needed in real data gathering, and it illustrates many important ideas in sampling. Another way of learning about sampling is to discuss surveys reported in the popular press. Reading and critiquing unusual news stories is fun, and real survey findings emphasize the importance of the statistical idea being illustrated. We have developed several demonstrations and examples of survey sampling to use in the classroom, and we use these activities regularly in our introductory statistics courses for college students who have completed high school algebra. We have also had success introducing the activities to eighth-grade students in New York City public schools, and since these activities are in the spirit of the NCTM's Standards for statistics for grades 9–12, they would be particularly useful in the high school setting.
Early Secondary Mathematics: The Use of Intrigue to Enhance Mathematical Thinking and Motivation in Beginning Algebra
Authors: Lewkowicz Marjorie L.
Published: November 2020

IN AN EFFORT TO HELP MY BEGINNING ALGEBRA STUDENTS further develop their conceptual understanding of variables, the language of algebra, and other important topics,
Media Clips: February 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.
My Calculator Is Broken; It Says the Log of (−1) Is …
Authors: Kahan Jeremy A. , Richgels Glen W.
Published: November 2020

Recently, a student in the mathematical methods for secondary teachers class that I (Richgels) teach stopped me in the hall and asked a question: “What does it mean when a calculator gives you a number for the log of negative one?” Immediately, smiling inside because I thought that I had caught a rookie, I asked the student to show me the calculator. I was impressed because the student had a new TI-86. The student entered–1. When he pressed the key, I expected an error message. Instead the calculator displayed (0, 1.36437635384). I did a double take and asked for the calculator. I had seen the student enter the correct expression but had to verify the result for myself. For almost thirty years, I had taught that the logarithm of a negative number is not defined.
Technology Tips: February 2003
Authors: Gosse Paul , Hollebrands Karen Flanagan
Published: November 2020

This month's column sees a return to dynamic geometry. Daniel Scher presents a “proof,” which he claims is rigorous, of why the midpoints of the sides of a quadrilateral, joined in order, produce a parallelogram. Do you agree? Our second tip is a worthwhile introductory activity for any group new to a particular technology. Susan Hvizdos challenges us with a TI-83 Plus Scavenger Hunt.
Delving Deeper: Match Making: Fitting Polynomials to Tables
Authors: Cuoco Al , Goldenberg E. Paul
Published: November 2020

Welcome to “Delving Deeper,” a new department in the Mathematics Teacher. The concept for this department has been evolving for several years, but the basic goal is to provide teachers with a forum for their own mathematical investigations. Do you have an idea in your mind or on your computer that is based on one of your mathematical explorations? Here is a chance to get your idea off your hard drive and into print. Submissions of up to ten single-spaced pages are welcomed. Articles written jointly with others (including students or mathematicians) are encouraged.
Study Groups: Professional Growth Through Collaboration
Authors: Arbaugh Fran
Published: November 2020

Professional development opportunities for mathematics teachers are abundant in the United States. School-and district-based workshops, college and university courses, summer institutes, and local, state, and national meetings for K–12 mathematics teachers all combine to provide numerous opportunities for professional growth. Individual teachers often return from these types of experiences with new activities to use in their classrooms and new ideas about teaching mathematics. What is often missing from many of these types of professional development experiences is the opportunity for teachers to build ongoing and collaborative learning relationships with mathematics teachers in their own school buildings.
Calendar: March 2003
Authors: Benson Steve
Published: November 2020

Problems 1–4 and problem 6 come from the 1990–1991 Mathcounts Handbook. Problems 5, 7, and 9–13 were part of the 1991–1992 Mathcounts national competition; problems 5 and 7 are from the Team Round; and problems 9–13 are from the Countdown Round. Problem 14 was adapted from the Houghton-Mifflin Summer Brain Teaser at http//:www.eduplace.com/math/brain/index.html. Problems 15–31 are from the New York Interscholastic Mathematics League Spring 1990 competition.
Sharing Teaching Ideas: The Oral Quiz
Authors: Bakal Jacqueline
Published: November 2020

This article shares an idea that I used on Fridays when I was teaching high school mathematics at an all-girls Catholic high school. I liked to give some form of assessment on Fridays because the students were always very “antsy” and talkative, and assessment was a good way to wrap up the work of the week. However, the lesson did not always lend itself to a written assessment on that day. For Fridays when I was not giving a written assessment, I devised my oral quiz. I usually used this activity once or twice during each marking period. I also plan to use this idea in the college-level developmental mathematics classes that I currently teach.
Unexpected Answers
Authors: Shultz Harris S. , Shultz Janice W. , Brown Richard G.
Published: November 2020

Mathematics educators should never lose sight of what attracted them to this profession. Certainly, one common thread would have to be the love of the subject, which in turn is often based on such attributes as its logic, succinctness, history, simplicity, and rigor. In addition, the sheer fun that a person can have with mathematics is often a significant factor. Such amusement can be encountered through questions whose answers seem to conflict with intuitive senses. One of the best-known examples of this type is the “birthday problem,” which asks for the probability that at least two people in a given room share the same birthday. Surprisingly, with just twenty-three people in the room, the probability that at least two have the same birthday is more than 50 percent. And with just forty people in the room, this probability is nearly 90 percent. Johnson (1997) includes a detailed discussion of the birthday problem, and Hathaway (2003) includes an interesting variation. This article examines other, perhaps less familiar, questions that have answers that many teachers and students will not expect. The reader is encouraged to read each problem and offer a response before reading the analysis.
Media Clips: May 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.
Mathematics Examinations: Russian Experiments
Authors: Karp Alexander
Published: November 2020

The goal of this article is to describe the objectives and methods of Russia's—more precisely, of St. Petersburg's—graduation examinations in high school mathematics. Although some interesting studies have described the experience of other nations (see, e.g., Dossey [1996]), the information is not widely disseminated and possible implications for American practice are not discussed much. However, the attention of both the public and the policy makers is now directed at the need for educational assessment, and other examination systems can serve as working experiments in methods of assessment. Such a comparison does not imply constructing a crude tabular comparison between systems of instruction and examination results or constructing simplistic hierarchies of teaching approaches, since curricula and the focus of teaching vary from one system to another. Our increased awareness of the full range of mathematics now being taught, in both content and pedagogy, should inform our own discussions of these issues. The Russian experience in teaching mathematics is a case in point, and knowledge of this experience might help anyone who is interested in teaching the subject.
Teaching Beginning Courses: Flower Power: Creating an Engaging Modeling Problem to Motivate Mathematics Students at an Alternative School
Authors: Koellner-Clark Karen , Newton Janice
Published: November 2020

Using rich problem activities that require modeling can be engaging for students who struggle to understand the content in entrylevel courses. Further, they provide students with a forum in which communication and problem solving are expected. According to Principles and Standards for School Mathematics, “Interacting with others offers opportunities for exchanging and reflecting on ideas; hence, communication is a fundamental element of mathematics learning” (NCTM 2000, p. 348).
My Journey toward a New Slant on Slope
Authors: Johnsen Connie , Wilkerson Trena L.
Published: November 2020

Learning how to modify my teaching style and determining successful methods for reaching struggling students in a beginning algebra class has been an eventful journey for me. My journey began fourteen years ago when I altered my teaching style of straight lecturing to one that incorporated handson activity learning with real-world connections, along with the lectures. In a beginning algebra class, using real-world connections is particularly important because the students have great difficulty connecting the abstract concepts of algebra with their everyday lives. Since research has shown that “50 percent of students sixteen and older function at Piaget's concrete operational level” (Gadanidis 1994, p. 93), high school students need the opportunity for hands-on activities. Businesses require that students not only comprehend but also apply mathematics to qualify for jobs.
Activities for Students: Stop This Runaway Truck, Please
Authors: Nelson Joanne E. , Coffey Margaret , Huffman Edie
Published: November 2020

Students are curious about the uses of mathematics in real life. Teachers are always looking for ideas for relating their curricula to their students' curiosity. The authors of this article met one summer at a National Security Agency workshop and developed this activity from a model of a problem that seemed interesting and enjoyable. Little did we know that the challenging mathematics that would develop from the truckescape-ramp problem would illustrate an application of mathematics that has saved countless lives.
Connecting Research to Teaching: Case Study: Advanced Placement Calculus in the Age of the Computer Algebra System
Authors: Dick Thomas P. , Dion Gloria S. , Wright Craig L.
Published: November 2020

The widespread availability of such powerful software as computer algebra systems (CASs) marks the latest chapter in the ongoing story of technological evolution and its implications for mathematics education. The existence of CASs should force us to think carefully and reexamine ideas regarding curriculum and instruction (what and how we teach) and assessment of student learning (what and how we evaluate). We can gain some insight from considering the experience of the College Board's Advanced Placement Program. Advanced Placement (AP) calculus can serve as a case study of a program that has had to come to grips with a variety of CAS-related issues in curriculum, instruction, and assessment.
Equal Arcs, Triangles, and Probability
Authors: Smith Richard J.
Published: November 2020

Typical high school mathematics courses too often postpone probability until the end of the year. Traditional textbooks relegate this area of mathematics to a section toward the end of the book, and teachers offer it to their students if time permits. I try to connect and integrate probability into the curriculum when appropriate.
Teaching Algebra to Students with Learning Disabilities
Authors: Steele Marcee M. , Steele John W.
Published: November 2020

In many states, students with learning disabilities are required to take algebra in high school. These students are usually served in inclusive settings (the general education classroom), where they often struggle with algebra because the content is so abstract. Although limited research and literature are available on algebra instruction for students with special needs, recognized strategies that promote students' learning can help make the algebra experience more enjoyable and successful for them and for the other students in the class, as well. This article highlights characteristics of students with learning disabilities, reviews current literature on algebra and students with learning disabilities, summarizes some of the recommendations, and describes how they can be put into practice.
Sharing Teaching Ideas: Algebra 1's Speed Trap
Authors: Gupta Neil Kenneth
Published: November 2020

When will i ever use this in life”? This question evoked many childhood memories of mathematics classes in which I struggled to learn the real-life applications of many of the concepts that I was learning: The question has also acted as a driving force to impel me to connect mathematical concepts with real-life situations. As a result, I devised a hands-on activity designed to engage learners. It used time-rate-distance problems to reinforce many mathematics concepts in an algebra 1 class. As a student teacher at Loveland Hurst Middle School in Loveland, Ohio, I developed this activity with my cooperating teacher. In this lesson, students measure the speed of moving cars, on the basis of the time spent traveling a given distance, and relay the information back to the drivers.
Is It Always True? From Detecting Patterns to Forming Conjectures to Constructing Proofs
Authors: Bremigan Elizabeth George
Published: November 2020

Suppose that a student comes to algebra class eager to share with her teacher and her classmates the following problem and its “amazing outcome.”
Using Research Projects to Help Develop High School Students' Statistical Thinking
Authors: Groth Randall E. , Powell Nancy N.
Published: November 2020

Statistics plays a key role in shaping policy in a democratic society, so statistical literacy is essential for all citizens to keep a democratic government strong (Wallman 1993). However, fostering the statistical thinking is a complex endeavor. We ultimately need to engage students in all phases of the investigative cycle of statistics, including data gathering, data analysis, and inference.
A Primer for Preproblem Ponderings: Anticipating the Answer
Authors: Cohen Karen Singer , Adams Thomasenia Lott
Published: November 2020

The value of anticipating the answer as a problem solving technique. Article discusses questions to ask about the form of an answer and its relationship to the conditions of the problem. Includes several examples useful for all students to develop a problem solving strategy.
Using Simple Quadratic Equations to Estimate Equilibrium Concentrations of an Acid
Authors: Brilleslyper Michael A.
Published: November 2020

Making meaningful connections between mathematics and other scientific disciplines has become a priority in education. In recent years, numerous national reports have called for more cross-disciplinary material in entry-level mathematics courses. (Cohen 1995 is one such example.) In this article, we consider an application of quadratic equations to a standard problem in chemistry. We show how quadratic equations arise naturally in solving the problem and then go on to consider a standard approximation using a simpler quadratic. This focus brings to light several important mathematical ideas. They include the sensitivity of the solutions on the equation coefficients and the use of inequalities to obtain upper bounds on the error
Media Clips: March 2004
Authors: Vennebush G. Patrick
Published: November 2020

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

The Millennium Mathematics Project (MMP) was set up within the University of Cambridge in 1999 as a joint project between the faculties of mathematics and education. It brings together a number of existing outreach activities, which have since been developed and extended and which now have a national and international user base. The MMP aims to support maths education and promote the development of mathematical skills and understanding, particularly through enrichment and extension activities beyond the school curriculum.
The Triangles of Aristarchus
Authors: Hirshfeld Alan W.
Published: November 2020

The ancient Greek mathematician Aristarchus demonstrated for the first time how it was possible, using simple observations and elementary geometry, to measure distances to bodies in the solar system. Aristarchus' methods used a lunar elcipse to approximate the diameter of the Earth, and used the shadow cone of a lunar eclipse to form similar triangles and proportional measurements. The mathmatics can be easily understood by a high school geometry student.
Tech Time: April 2004
Authors: Hollebrands Karen Flanagan , Stohl Hollylynne
Published: November 2020

AS MANY OF YOU PREPARE YOUR STUDENTS FOR ENDof-year examinations, you spend time creating review sheets and sample assessments. For this month's “Technology Tip,” Jeff Suzuki introduces readers to a powerful use of word processing and spreadsheet software that can help teachers design individualized worksheets and assessments. Jeff provides directions for using a function called Mail Merge in Microsoft Office. Although most productivity software groups (for example, ClarisWorks, Microsoft Works, and Office XP) have this type of merging function, the directions given in this “Technology Tip” are specific to Microsoft Office 2000. The process may be used with any software that has mail-merge capabilities and that allows a spreadsheet data file as input.
Projects: Mathematics Online Projects Connect Students Worldwide
Authors: Foletta Gina
Published: November 2020

Mention “collaborative learning,” and teachers might envision elementary school students working together on an interdisciplinary lesson, but high school students? Some believe that collaborative projects are best left for younger students.
Sharing Teaching Ideas: The Extension-Reduction Strategy: Activating Prior Knowledge
Authors: Sloyer Clifford W.
Published: November 2020

In a beginning calculus class consisting of firstyear college students, an assigned problem required students to find the volume of a cup with unequal radii at the top and bottom. Not one student in the class offered any ideas directed toward solving this problem.
Tech Time: August 2004
Authors: Stohl Hollylynne , Harper Suzanne R.
Published: November 2020

Welcome to a new year of “Technology Tips.” I, Hollylynne Stohl, will edit the column for the 2004–2005 year. If you have a tip that can help other teachers learn how to use a technology application in the classroom, please send your ideas to me.
War and Pieces
Authors: Hansen Will
Published: November 2020

Tolstoy's analogy of the correct interpretation of history as similar to following the laws of math and physics. Provides lesson plans and alternatives for calculus students studying intervals and integration. An interdisciplinary approach to integrals.
Delving Deeper: A Specific Construction of a Conic From An Ellipse
Authors: Lee J. Todd
Published: November 2020

Consider the interesting geometric construction given by Manuel Santos-Trigo in the “Technology Tips” in the January 2004 Mathematics Teacher (Santos-Trigo 2004). He starts with an ellipse, its center point O, a variable point R on the line along the major axis of the ellipse, and variable points S and S', which are points on the ellipse that are reflections with respect to the major axis. Figure 1 shows the general setup and various placements of S. Santos-Trigo constructs a pair of lines, one through R and S and the other through O and S'. He describes a set of discovery exercises involving the locus of points generated by the intersection of these lines as the point S roams around the ellipse. The empirical conclusion of these exercises is that the locus is a conic section, the nature of which is determined in a fairly simple way by the location of R relative to O and the major vertices of the ellipse.
Patterns Jumping Out of a Simple Checker Puzzle
Authors: Staples Susan G.
Published: November 2020

In this article, we consider the generalization of a “solitaire checker puzzle” from the book More Joy of Mathematics, by Theoni Pappas (1991). In addition to presenting the solution to the general case, we shall also investigate the attractive patterns that emerge during the process of solving the puzzle, as well as in analyzing the minimal solutions of various cases.
Fractal Patterns and Chaos Games
Authors: Devaney Robert L.
Published: November 2020

One of the most wonderful ways to introduce students in middle school or secondary school to the beauty and excitement of contemporary mathematics is to involve them in the many variations of the “chaos game” which produces such intriguing fractal patterns as the Sierpinski triangle and the Koch curve.
Making Mathematical Connections by Constructing Tetrahedra
Authors: Bergner Jennifer A. , Groth Randall E.
Published: November 2020

Part of the beauty of mathematics is that seemingly isolated branches of the subject can often be used together to produce solutions to problems. High school students need to engage in activities that help them see how the various branches of mathematics work together in problem-solving situations. NCTM (2000) underscores the importance of such activities, stating, “When students can see the connections across different mathematical content areas, they develop a view of mathematics as an integrated whole” (NCTM 2000, p. 354).
Activities for Students: Curve Stitching: Linking Linear and Quadratic Functions
Authors: Bergthold Trisha
Published: November 2020

Curve stitching is the craft of creating the illusion of a curve by stitching a sequence of strategically placed straight stitches. Invented in the mid–1800s by Mary Boole, this craft allows artistic expression and mathematical activity at several different levels. A child as young as four or five years old can complete easy designs, yet curve stitching can also be sophisticated enough to create very beautiful art. Over the past century, sculptors Naum Gabo, Antoine Pevsner, Henry Moore, and Barbara Hepworth have made extensive use of curve stitching in their sculptures.
Graphing Portfolios in Calculus: Reinforcing Concepts and Inviting Creativity
Authors: Bailey Evelyn C. , Chen Fang
Published: November 2020

In this article, we introduce the idea of a graphing portfolio and describe its implementation in two levels of otherwise traditional university calculus courses. We believe that graphing portfolios facilitate learning because they require that students work backward from the traditional function to a graph sequence: To create a graphing portfolio, students must find functions whose graphs have specific shapes and domains. This process gives students the opportunity to investigate a variety of functions and transformations while incorporating their artistic and cultural strengths. The graphing portfolio is assigned after the class completes a unit on curve sketching as a way to apply concepts of calculus. We have found that this activity helps students gain a broader and deeper understanding of the connection between a function and its graph.
Promoting Understanding of Linear Equations with the Median Slope Algorithm
Authors: Edwards Michael Todd
Published: November 2020

In recent years, the influence of technology on school mathematics curricula has become increasingly pronounced. Today, for instance, calculators seem ever–present in most mathematics classrooms. In fact, a recent study of nearly 4,000 mathematics teachers and 50,000 students reports that approximately 96 percent of eighth–grade students in the United States have access to calculators for use in mathematics classrooms (IEA 2001). The computational power of handheld calculators—particularly those equipped with graphing capabilities—has significantly accelerated the introduction of numerous mathematical topics within school mathematics. For example, before the early 1990s, leastsquares regression was not found in most algebra textbooks. Because by–hand methods for calculating a least–squares regression line required students to minimize the sum of squared residuals—a technique requiring differentiation—the procedure was rarely studied before calculus. Now, however, any student with a graphing calculator can perform a leastsquares linear regression by pressing only a few buttons. Not surprisingly, linear regression is now commonly found in the school algebra curriculum.
Delving Deeper: Reducing the Sum of Two Fractions
Authors: Shultz Harris S. , Shiflett Ray C.
Published: November 2020

Sometimes when we add fractions, the sum can be reduced even though we have used the least common denominator. Other times, the sum cannot be reduced. For example, if we add 1/3 + 1/6, the sum 3/6 can be reduced to 1/2. However, 2/5 + 1/6 = 17/30 cannot be reduced.
Activities for Students: Using Overhead Projectors to Explore Size Change Transformations
Authors: Edwards Michael Todd
Published: November 2020

Through the study of transformation, a wide range of mathematical concepts may be introduced to secondary school students.
Cribbage: An Excellent Exercise in Combinatorial Thinking
Authors: Markel William D.
Published: November 2020

Card games have long been a rich source of combinatorial exercises. Indeed, determining the probabilities of obtaining various hands in poker, and often in bridge, has been standard fare for elementary texts in both probability and combinatorics. Examples involving the game of cribbage, however, seem rare. This omission is especially surprising when one considers that cribbage hands offer excellent applications of combinatorial reasoning.
Promoting Problem Solving across Geometry and Algebra by Using Technology
Authors: Erbas A. Kursat , Ledford Sarah D. , Orrill Chandra Hawley , Polly Drew
Published: November 2020

Technology is a powerful tool in assisting students in problem solving by allowing for multiple representations. It is particularly effective when students are encouraged to explore a single mathematics problem with multiple technologies. Using technology allows students to be creative in their problem-solving techniques, look at data in a variety of ways, and see the viability of their answers, all of which may lead to a better understanding of mathematics. The vignette offered in this article provides insight into ways to solve open-ended problems by using multiple technologies. Although the series of events described is hypothetical, the original problem is familiar and the authors' collective experiences with such tasks and high school students indicates that the students are able to explore problems in ways that demonstrate and extend their mathematical thinking when they have appropriate teacher support.
Activities for Students: Averaging Rates: Deciding When to Use the Harmonic or Arithmetic Mean
Authors: Brown S. L. , Rizzardi M. A.
Published: November 2020

When asked to calculate an average, virtually everybody assumes the calculation consists of adding up all the values and dividing the total by the number of values.
Data-Driven Mathematics Investigations on Curved Data
Authors: Bossé Michael J.
Published: November 2020

Investigations of real–world data begin in elementary school. Students often produce scatter plots, leading to trend lines. In the middle grades, lines of best fit are often investigated through median–median lines and double–centroid lines (Shawer et al. 2002). In the secondary grades, linear regression is produced by the least squares line. While these techniques are adequate for data that is more or less linear, teachers and students often encounter data that produce a “curved” scatter plot. In these cases additional techniques are required. This article demonstrates three techniques to determine the equation of a polynomial function through two or more points that model the graph of “good fit” for a set of data. Using these techniques, students can develop functions through which they can evaluate mathematical behavior and make predictions. Secondary mathematics teachers will find these techniques particularly valuable. Each technique can be applied within various secondary mathematics courses such as algebra 2, statistics, or precalculus.
Sharing Teaching Ideas: Teaching Logarithms Day One
Authors: Gamble Marvin
Published: November 2020

Students being taught logarithms for the first time often memorize the exponent and logarithm properties and never understand the material. Teachers who have taught this concept have repeatedly told the students that logarithms are exponents. For some reason students hear the terms exponents and logarithms but often do not understand the relationship between them. I have had a great deal of success in helping students understand the relationship between exponents and logarithms in high school and college courses using the following procedure.
Connecting Research to Teaching: Enhancing Meaning in Mathematics: Drawing on What Students Know about the Physical World
Authors: Marrongelle Karen A.
Published: November 2020

Imagine your students working on the task shown in figure 1. Before reading on, think about the next two questions: How might your mathematics students make sense of this task? What do you think your students will need to attend to in order to successfully explore this task?
Tapping into Trapezoids
Authors: Wanko Jeffrey J.
Published: November 2020

The use of trapezoids to explore a number of mathematical concepts, including similarity, representation, and the Pythagorean theorem. Preservice teachers develop hypotheses about isosceles trapezoids which are investigated. Tiling with pattern blocks and the development of the formula for area are also examined.