## Classroom-Ready Resources and Teaching Suggestions

Browse index of informative background articles for college algebra, pre-calculus, and trigonometry.

Return to master index.

##### Conic Sections

Mathematical Mysteries of Rapa Nui with Classroom Activities, by Ximena Catepillán, Cynthia Huffman, and Scott Thuong

Mathematical mysteries of Rapa Nui (Easter Island) and related classroom activities, based on archeological findings from a recent field trip. Activity topics include conic sections, equation of ellipses, and curve fitting. Also available in Spanish translation as Misterios Matemáticos de Rapa Nui con Actividades para el Aula de Clases.

##### Determinants

Analysis and Translation of Raffaele Rubini's 1857 'Application of the Theory of Determinants: Note', by Salvatore J. Petrilli, Jr., and Nicole Smolensk

A compendium of early determinant theory offered in defense of “analytic” mathematics, with suggestions for its use in pre-calculus and linear algebra courses.

##### Exponentials and Logarithms

Logarithms: The Early History of a Familiar Function, by Kathleen M. Clark and Clemency Montelle

A recounting of Napier's and Burgi's parallel development of logarithms told as a ‘great tale’ for use in the classroom. Includes student exercises.

##### Polynomials and Roots

Completing the Square: From the Roots of Algebra, A Mini-Primary Source Project for Students of Algebra and Their Teachers, by Daniel E. Otero

One of a collection of student-ready modules based on primary historical sources presented in the article A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources.

Mark Kac’s First Publication: A Translation of “O nowym sposobie rozwiązywania równań stopnia trzeciego”, by David Derbes

English translation of Mark Kac's first publication on a new derivation of Cardano’s formula, written while he was still in high school, with a typescript of the original Polish article, a biographical synopsis of Kac, the tale of the rediscoveries of the paper, and suggestions for classroom discussions of the cubic.

Solving the Cubic with Cardano, by William B. Branson

The author shows how, in order to solve the cubic, Cardano relied on both classical Greek geometry and *abbaco* traditions. He illustrates Cardano’s geometric thinking with modern manipulatives.

To Simplify, or Not To Simplify? A Lesson from Medieval Iraq, by Valerio De Angelis and Jeffrey A. Oaks

A case where not simplifying fractions explains a curious rule for computing cube roots from medieval Arabic mathematics accompanied by a set of student exercises based on that rule.

##### Trigonometry

Misseri-Calendar: A Calendar Embedded in Icelandic Nature, Society, and Culture, by Kristín Bjarnadóttir

History of this two-season calendar from Viking times to today, with animations and ideas for your classroom.

Poles, Parking Lots, & Mount Piton: Classroom Activities that Combine Astronomy, History, and Mathematics, by Seán P. Madden, Jocelyne M. Comstock, and James P. Downing.

Measurement activities based on historical methods that combine basic ideas from high school geometry and trigonometry with astronomical data that students can collect themselves.

Teaching and Learning the Trigonometric Functions through Their Origins, by Daniel E. Otero

Collection of six classroom-ready curricular units based on primary historical sources designed to serve students as an introduction to the study of trigonometry.

Van Schooten's Ruler Constructions, by C. Edward Sandifer

With suggested exercises on trigonometry.

## Informative Background Articles

Browse index of classroom-ready resources and teaching suggestions for college algebra, pre-calculus, and trigonometry.

Return to master index.

##### General Algebra

A Classic from China: The Nine Chapters, by Randy K. Schwartzo demonstrate some of the intuitive techniques employed by the Chinese in establishing their proofs: the use of grids or squares to comprehend area and their dissection techniques to illustrate algebraic operations. History of and problems for students from this early and influential Chinese work.

The ‘Piling Up of Squares’ in Ancient China, by Frank Swetz

Intuitive geometric techniques employed by the Chinese in establishing their proofs: the use of grids or squares to comprehend area and their dissection techniques to illustrate algebraic operations.

The Unique Effects of Including History in College Algebra, by G. W. Hagerty, S. Smith, and D. Goodwin

Using the history of mathematics in a college algebra class has had significant positive effects on student learning.

##### Polynomials and Roots

A Modern Vision of the Work of Cardano and Ferrari on Quartics, by Harald Helfgott and Michel Helfgott

A study of the solution of quartic equations in Cardano's *Ars Magna* and in the work of Euler and Descartes.

An Arabic Finger-reckoning Rule Appropriated for Proofs in Algebra, by Jeffrey A. Oaks

In a 1301 work, Ibn al-Bannāʾ based his proofs on a common mental multiplication technique.

Completing the Square, by Barnabas Hughes

Explain the geometric basis of “completing the square,” the original method of solving quadratic equations, to your students.

Descartes’ Method for Constructing Roots of Polynomials with ‘Simple’ Curves, by Gary Rubinstein

Descartes' methods from his 1637 ‘Geometry’ explicated and illustrated using interactive applets.

Geometric Approaches to Quadratic Equations from Other Times and Places, by Patricia R. Allaire and Robert E. Bradley

A historical survey of geometric techniques for solving quadratic equations from ancient Babylonia, classical Greece, medieval Arabia, and early modern Europe that can enhance an algebra course.

How Tartaglia Solved the Cubic Equation, by Friedrich Katscher

The method of Tartaglia for solving cubics, which he eventually explained to Cardano.

##### Exponentials and Logarithms

Celebrating a Mathematical Miracle: Logarithms Turn 400, by Glen Van Brummelen

Why John Napier's invention of logarithms in 1614 was hailed as a miracle by astronomers and mathematicians.

Napier's *e*, by Amy Shell-Gellasch

A light-hearted trip back through time to when *e* was young that looks at the question of why we use *‘e*’ to represent the base of the natural logarithm system.

##### Trigonometry

Historical Reflections on Teaching Trigonometry, by David M. Bressoud

An argument for beginning the study of trigonometry with the circle definitions of the trigonometric functions and angle measurements, based on its historical development.

Triangles in the Sky: Trigonometry and Early Theories of Planetary Motion, by Sandra M. Caravella

Survey of early theories of planetary motion, with dynamic figures to help in understanding these motions.

Return to master index.