Completing the Square: From the Roots of Algebra

This mini-PSP is designed to give students a deep understanding of the method of completing the square, to serve as a bridge between the method of factoring and the quadratic formula, all of which are techniques considered by all students of beginning algebra. For such students, completing the square can involve procedural steps that mysteriously produce the required sought-for answers, and the quadratic formula can act like a runic talisman that magically generates the right numbers that solve the given equation. A coherent treatment that emphasizes a reasoned approach to these topics is quite naturally found in the history of the development of this method. Students learn the method of completing the square from the “Father of Algebra,” Muḥammad ibn Mūsā al-Khwārizmī, a ninth-century scholar from the then-young city of Baghdad. Written about the year 825, al-Khwārizmī ‘s extremely influential work on the subject, with the title al-Kitāb al-mukhtaṣar fī hisāb al-jabr wal-muqābala (The Compendious Book on Calculation by Restoration and Reduction), better known today simply as Algebra, instructs his readers how to find the roots of a quadratic equation. But al-Khwārizmī’s equations are ones without symbols and are expressed entirely in words. This rhetorical algebra of al-Khwārizmī provides a careful description of the method we call completing the square, along with a clear geometric demonstration of how the method works that involves completing a real (geometric) square!

Greatest Common Divisor: Algorithm and Proof

The greatest common divisor of two numbers is a fundamental concept in number theory, and also appears in discrete math and ring theory in abstract algebra. The ancient Chinese and ancient Greeks independently discovered similar algorithms for finding the greatest common divisor. The Chinese presented their algorithm in the context of simplifying fractions, and justified it by a very concrete argument. Euclid presented his algorithm in his book on number theory and proved it with a careful logical proof. This PSP explores both algorithms and the context in which they are presented. It also presents the algorithm in modern notation and compares the proofs from the ancient Chinese, Euclid and a careful modern proof. This small slice of the history of proof would also make a great lesson in an introduction to proof course.

Euler’s Calculation of the Sum of the Reciprocals of the Squares

In this project, we guide students through Euler’s incredibly clever 1740 solution of the Basel Problem, in which he proved that the series of the reciprocals of positive integer squares converges to π2/6. Although issues related to series convergence were viewed differently in the 18th century, today’s standard series convergence tests are used heavily throughout the project. No prerequisite knowledge is required to understand the proof itself beyond the power series for sine and basics from precalculus such as finding zeros of functions and factoring polynomials. While this project is intended for an introductory calculus course, it also makes an ideal starting point for a discussion of the Riemann zeta function or the Weierstrass Factorization Theorem in a complex analysis course, or a discussion of generating functions in a combinatorics course.

The Closure Operation as the Foundation of Topology

The axioms for a topology are now well established: closure under unions of open sets, closure under finite intersections of open sets, and the fact that the entire space and empty set are open. However, in the early twentieth century, multiple systems were proposed as options for defining a topology. Once such system was based on the closure property, and it was the subject of Polish mathematician K. Kuratowski’s doctoral thesis. In this mini-project, students work through a proof that today’s axioms for a topology are equivalent to Kuratowski’s closure axioms by studying excerpts from works by both Kuratowski and Hausdorff. We will also discuss ideas for how to use this project as a jumping-off point for an exploratory “build your own topology” project, similar to how students sometimes build their own geometry in a geometry course.

Summer Tour of TRIUMPHS PSPs

This webinar introduces a subset of the TRIUMPHS PSP collection, and gives enough description of each to help participants decide which they want to use during the 2019-2020 academic year. The session also contains some information about how to become an official TRIUMPHS site tester.

A Genetic Context for Understanding the Trigonometric functions

This webinar highlights Danny Otero’s PSP titled A Genetic Context for Understanding the Trigonometric Functions.  This project, designed for use by students in a Trigonometry or Precalculus course, aims to provide a context for study of this subject. The project is also ideal for use in a History of Mathematics course, as it takes sweeping thematic perspective on a particular topic. It addresses the questions, What is trigonometry? Where did it come from? And what are its main ideas? The project provides an overview of the genesis of modern trigonometry through six brief vignettes: measuring angles through Babylonian numeration; a reconstruction of Hipparchus’ table of chords; Ptolemy’s solution of a simple problem in mathematical astronomy; Sanskrit verse of Varahamihira which contains a “code” for building a table of sines; al-Biruni’s work with measuring shadows; and an early comprehensive account of trigonometry in the West by Regiomontanus.
The PSP can be downloaded at https://digitalcommons.ursinus.edu/triumphs_precalc/1/

Seeing and Understanding Data

This webinar highlights the project Seeing and Understanding Data, authored by  Charlotte Bolch and  Beverly Wood. The primary focus of the mini-PSP Seeing & Understanding Data is fostering student appreciation for the power of data displays as they have evolved from hand-drawn sketches to dynamic web apps. Students are given the opportunity to explore the evolution of statistical graphs and visual displays across time from William Playfair’s graphs to the interactive display on Gapminder as well as think critically about how data are displayed and interpreted. The PSP can be download at https://digitalcommons.ursinus.edu/triumphs_statistics/2/

Babylonian Numeration

This webinar highlights the project Babylonian Numeration, authored by  Dominic Klyve.   Rather than being taught a different system of numeration, students in this project discover one for themselves. Students are given an accuracy recreation of a cuneiform tablet from Nippur with no initial introduction to Babylonian numerals. Unknown to the students, the table contains some simple mathematics – a list of the first 13 integers and their squares. Their challenge is threefold: 1) to deduce how the numerals represent values, to work out the mathematics on the tablet, and to decide how to write `”72”. A small optional extension of the project asks students to compare the good and bad traits of several numeration systems.  The PSP can be download at https://digitalcommons.ursinus.edu/triumphs_number/2/