Mathematical Modeling

Applications with GeoGebra

Jonas Hall, Thomas Lingefjärd

• Éditeur: Wiley
• Année: 2016
• ISBN:
  9781119102847
• Édition: 1
• Sujet: Mathematics
• Sous-sujet: Mathematics General

A logical problem-based introduction to the use of GeoGebra for mathematical modeling and problem solving within various areas of mathematics

A well-organized guide to mathematical modeling techniques for evaluating and solving problems in the diverse field of mathematics, Mathematical Modeling: Applications with GeoGebra presents a unique approach to software applications in GeoGebra and WolframAlpha. The software is well suited for modeling problems in numerous areas of mathematics including algebra, symbolic algebra, dynamic geometry, three-dimensional geometry, and statistics. Featuring detailed information on how GeoGebra can be used as a guide to mathematical modeling, the book provides comprehensive modeling examples that correspond to different levels of mathematical experience, from simple linear relations to differential equations.

Each chapter builds on the previous chapter with practical examples in order to illustrate the mathematical modeling skills necessary for problem solving. Addressing methods for evaluating models including relative error, correlation, square sum of errors, regression, and confidence interval, Mathematical Modeling: Applications with GeoGebra also includes:

• Over 400 diagrams and 300 GeoGebra examples with practical approaches to mathematical modeling that help the reader develop a full understanding of the content
• Numerous real-world exercises with solutions to help readers learn mathematical modeling techniques
• A companion website with GeoGebra constructions and screencasts

Mathematical Modeling: Applications with GeoGebra is ideal for upper-undergraduate and graduate-level courses in mathematical modeling, applied mathematics, modeling and simulation, operations research, and optimization. The book is also an excellent reference for undergraduate and high school instructors in mathematics.

1
SOME INTRODUCTORY PROBLEMS

The introductory problems in this chapter focus on developing students’ abilities to reason and produce a reasonable model from a real‐world situation. Photographs are used in several examples and the student may easily expand the problem assignments by providing their own photographs and related questions. The problems are not necessarily “beginners” problems; they are selected because they are “nonstandard,” and offer a glimpse into some of the different possibilities of modeling with GeoGebra. The problems presented in the next chapter are more standard and may for some beginners be a better choice.

Students should first try to solve some part of a problem “by head and hand,” using a calculator and paper and pencil and then expand to the other parts using GeoGebra. To learn mathematics in the present digital age, we need to recognize that old school content and methods have been overtaken by a digital technology that has moved us into a mathematical golden age. To be effective in teaching mathematics with technology‐assisted instruction, we suggest that you study the TPACK framework and work toward an integrated approach where technology, mathematical content, and pedagogy are all integrated seamlessly in every lesson.

We begin with a couple of different introductory problems. As students enter their upper secondary mathematical studies, the core content suddenly needs to be mastered in greater depth than previously. Some examples include, but are not limited, to the following:

• Understanding numbers, arithmetic, and algebra.
• Properties of a range of whole numbers, different number bases, and the concepts of prime numbers and divisibility.
• Calculations in everyday life and with real numbers, written in different forms, including powers with real exponents, together with strategies for calculations when using digital tools.
• Generalization of the rules of arithmetic to handle algebraic expressions. The concept of linear inequality.
• Algebraic and graphical methods for solving linear equations and inequalities, and exponential equations.
• Concepts of sine, cosine, and tangent, and methods of calculating angles and lengths of right‐angled triangles. The concept of a vector and its representations, such as direction, length, and points in a coordinate system.
• Addition and subtraction with vectors and scalar multiplication to produce a vector.
• Mathematical reasoning using basic logic, including implication and equivalence, and comparisons with how to argue in everyday contexts and in science subjects.
• Illustration of the concepts of definition, theorem, and proof, such as the Pythagorean Theorem and the sum of the angles of a triangle.
• Advanced percentage concepts: ppm and percentage points.
• Concepts of rate of change and indexes, as well as methods for calculating interest and mortgage payments for different types of loans.
• Concept of a function, its domain and range of a definition, and also properties of linear functions, exponential, and polynomial functions.
• Represent...