- Description
- Resource Material
The course begins with the mathematical theory of linear Diophantine equations. The math teacher recalls basic number theory (from 5th grade) –divisibility rules and the Euclidean algorithm for finding the greatest common divisor (gcd) of two positive integers. The math teacher can begin introducing the new material by posing to the students a simple everyday life problem which can be modeled by a linear non-homogenous Diophantine equation of two variables. The teacher guides the students in creating the model and finding the solutions to the obtained equation. The students use divisibility rules to solve the equation. Then, the math teacher introduces the students to the term “Diophantine equation”, in particular to linear Diophantine equations of two variables (ax + by = c) and underlines the differences between a linear equation of one variable and an undefined multivariable equation. The math teacher introduces the students to basic methods for solving linear non-homogenous Diophantine equations: by using divisibility rules, by using the extended Euclidean algorithm for finding a particular solution and then writing the formulas for the general solution, and by the Euler’s substitution method. Discusses with the students the necessary and sufficient condition for such an equation to have integer solutions, the number of integer solutions and the number of solutions in natural numbers (constrains to the variables expressed by linear inequalities). The math teacher shows also how to solve a linear Diophantine equation of more than two variables. The math teacher also introduces the students to the Frobenius coin problem and its generalizations. To the more advanced and curious students, the math teacher can explain the relation between linear Diophantine equations and congruencies, i.e. how such an equation can be expressed in the form of a linear congruence. The math teacher may also explain how linear Diophantine equations can be applied to simple problems in cryptography.
Next, the IT teacher introduces the students to the methods which can be used for solving linear non-homogenous Diophantine equations in Excel. The chemistry teacher explains to the students how to balance a chemical equation which expresses a chemical reaction, and together with the math teacher explains how to apply linear non-homogenous Diophantine equations for balancing a chemical equation.
The physics (engineering) teacher introduces the students to networks and network flows. Optionally, the school administration can invite a (network) engineer to introduce students to networks and problems involving network flow. The math teacher and the physics teacher explain to the students how linear Diophantine equations can be used for modeling a flow in a network. The entrepreneurship teacher poses to the students business problems which can be modeled with linear non-homogenous Diophantine equations. The students can solve the obtained equations either by hand using the methods from math classes, or in a spreadsheet software.
The work on the subject lasts 10 hours.
