Exploring the fundamental theorem of arithmetic in excel 2007

This paper discusses how fundamentals of number theory, such as unique prime factorization and greatest common divisor can be made accessible to secondary school students through spreadsheets. In addition, the three basic multiplicative functions of number theory are defined and illustrated through a spreadsheet environment. Primes are defined simply as those natural numbers with just two divisors. One focus of the paper is to show the ease with which spreadsheets can be used to introduce students to some basics of elementary number theory. Complete instructions are given to build a spreadsheet to enable the user to input a positive integer, either with a slider or manually, and see the prime decomposition. The spreadsheet environment allows students to observe patterns, gain structural insight, form and test conjectures, and solve problems in elementary number theory.

Basic finance made accessible in Excel 2007 : "the big 5, plus 2"

The basic principles and equations are developed for elementary finance, based on the concept of compound interest. The five quantities of interest in such problems are present value, future value, amount of periodic payment, number of periods and the rate of interest per period. We consider three distinct means of computing each of these five quantities in Excel 2007: (i) use of algebraic equations, (ii) by recursive schedule and the Goal Seek facility, and (iii) use of Excel's intrinsic financial functions. The paper is intended to be used as the basis for a lesson plan and contains many examples and solved problems. Comment is made regarding the relative difficulty of each approach, and a prominent theme is the systematic use of more than one method to increase student understanding and build confidence in the answer obtained. Full instructions to build each type of model are given and a complete set of examples and solutions may be downloaded (Examples.xlsx and Solutions.xlsx).

Insight into the fractional calculus via a spreadsheet

Many students of calculus are not aware that the calculus they have learned is a special case (integer order) of fractional calculus. Fractional calculus is the study of arbitrary order derivatives and integrals and their applications. The article begins by stating a naive question from a student in a paper by Larson (1974) and establishes, for polynomials and exponential functions, that they can be deformed into their derivative using the μ-th order fractional derivatives for 0<μ<1. Through the power of Excel we illustrate the continuous deformations dynamically through conditional formatting. Some applications are discussed and a connection made to mathematics education.

Revisiting Polya's summation techniques using a spreadsheet : from addition tables to Bernoulli polynomials

Recurrence relations in mathematics form a very powerful and compact way of looking at a wide range of relationships. Traditionally, the concept of recurrence has often been a difficult one for the secondary teacher to convey to students. Closely related to the powerful proof technique of mathematical induction, recurrences are able to capture many relationships in formulas much simpler than so-called direct or closed formulas. In computer science, recursive coding often has a similar compactness property, and, perhaps not surprisingly, suffers from similar problems in the classroom as recurrences: the students often find both the basic concepts and practicalities elusive. Using models designed to illuminate the relevant principles for the students, we offer a range of examples which use the modern spreadsheet environment to powerfully illustrate the great expressive and computational power of recurrences.

Aspects of elementary number theory illustrated in the spreadsheet environment

Number theory has in recent decades assumed a great practical importance, due primarily to its application to cryptography. This chapter discusses how elementary concepts of number theory may be illuminated and made accessible to upper secondary school students via appropriate spreadsheet models. In such environments, students can observe patterns, gain structural insight, form and test conjectures, and solve problems. The chapter begins by reviewing literature on the use of spreadsheets in general and the use of spreadsheets in number theory in particular. Two sample applications are then discussed. The first, factoring factorials, is presented and instructions are given to construct a model in Excel 2007. The second application, the RSA cryptosystem, is included because of its importance to Science, Technology, Engineering, and Mathematics (STEM) students. Number theoretic concepts relevant to RSA are discussed, and an outline of RSA. is given, with example. The chapter ends with instructions on how to construct a simple spreadsheet illustrating RSA.

Spreadsheet Conditional Formatting Illuminates Investigations into Modular Arithmetic

Modular arithmetic has often been regarded as something of a mathematical curiosity, at least by those unfamiliar with its importance to both abstract algebra and number theory, and with its numerous applications. However, with the ubiquity of fast digital computers, and the need for reliable digital security systems such as RSA, this important branch of mathematics is now considered essential knowledge for many professionals. Indeed, computer arithmetic itself is, ipso facto, modular. This chapter describes how the modern graphical spreadsheet may be used to clearly illustrate the basics of modular arithmetic, and to solve certain classes of problems. Students may then gain structural insight and the foundations laid for applications to such areas as hashing, random number generation, and public-key cryptography.

Scientists in Control: a progressive approach to preparing work ready graduates in QC, EQA and method evaluation

Introduction QC, EQA and method evaluation are integral to delivery of quality patient results. To ensure QUT graduates have a solid grounding in these key areas of practice, a theory-to-practice approach is used to progressively develop and consolidate these skills. Methods Using a BCG assay for serum albumin, each student undertakes an eight week project analysing two levels of QC alongside ‘patient’ samples. Results are assessed using both single rules and Multirules. Concomitantly with the QC analyses, an EQA project is undertaken; students analyse two EQA samples, twice in the semester. Results are submitted using cloud software and data for the full ‘peer group’ returned to students in spreadsheets and incomplete Youden plots. Youden plots are completed with target values and calculated ALP values and analysed for ‘lab’ and method performance. The method has a low-level positive bias, which leads to the need to investigate an alternative method. Building directly on the EQA of the first project and using the scenario of a lab that services renal patients, students undertake a method validation comparing BCP and BCG assays in another eight-week project. Precision and patient comparison studies allow students to assess whether the BCP method addresses the proportional bias of the BCG method and overall is a ‘better’ alternative method for analysing serum albumin, accounting for pragmatic factors, such as cost, as well as performance characteristics. Results Students develop understanding of the purpose and importance of QC and EQA in delivering quality results, the need to optimise testing to deliver quality results and importantly, a working knowledge of the analyses that go into ensuring this quality. In parallel to developing these key workplace competencies, students become confident, competent practitioners, able to pipette accurately and precisely and organise themselves in a busy, time pressured work environment.

A preliminary investigation into technology and processes facilitating the assurance of learning

This paper reports on the outcomes from a preliminary evaluation of technologies and processes intended to support the Assurance of Learning initiative in the business faculty of an Australian university. The study investigated how existing institutional information systems and operational processes could be used to support direct measures of student learning and the attainment of intended learning goals. The levels at which learning outcomes had been attained were extracted from the University Learning Management System (LMS), based on rubric data for three assessments in two units. Spreadsheets were used to link rubric criteria to the learning goals associated with the assessments as identified in a previous curriculum mapping exercise, and to aggregate the outcomes. Recommendations arising from this preliminary study are made to inform a more comprehensive pilot based on this approach, and manage the quality of student learning experiences in the context of existing processes and reporting structures.