Volume III , issue 2 ( 2000 ) | back |

Selected chapters from algebra, IV. Primes | 63$-$82 |

**Abstract**

This paper is the fourth part of the publication ``Selected chapters from algebra'', the first three having been published in previous issues of the Teaching of Mathematics, Vol\. I (1998), 1--22, Vol\. II, 1 (1999), 1--30, Vol\. II, 2 (1999), 65--80, and Vol\. III, 1 (2000), 15--40.

**Keywords:** Primes, function $\pi(n)$, Chebyshev
inequality, asymptotical law of distribution of primes.

**MSC Subject Classification:** 00A35

How to motivate 10--18 years old pupils to work independentle on solving mathematical problems | 83$-$94 |

**Abstract**

The purpose of this article is to suggest a few problems for exploration which could motivate pupils to think and work independently on their solution. The article consists of three parts, describing experinces with pupils of different age groups: Part I describes a problem, related to solving the equation $ax+by=n$, with $a,b,nın N$ in non-negative integers $x$ and~$y$. This problem has been studied with pupils aged 14--15. Part II is devoted to sequences and combinatorial investigations of problems designed for 16--18 years old youngsters. In Part III the importance of introducing youngsters at an early age to independent work is underlined. This will be done by providing problems for children aged 10--13.

**MSC Subject Classification:** 00A35

Functions for fourth graders | 95$-$104 |

**Abstract**

Ordering is an important activity for cognitive development of a child and it has a fruitful form in the idea of (finite) sequence. A schematic representation of concrete sequences (of numbers or letters) is used to foster the idea of function. Leading pupils to discover simple laws of correspondence and to compose the expressions which formulate them seems to be the first possible instance of such activities feasible in elementary school practice.

**Keywords:** Sequences, discovering of rules, finding sums.

**MSC Subject Classification:** 00A35

The instruction of Euclidean geometry with the use of educational software in multilingual classes | 105$-$113 |

**Abstract**

In this article we aim to describe how the use of computer visualization software can assist the teaching of Euclidean geometry in a multilingual classroom. In particular, the difficulties that the students may experience are presented, together with some advantages and disadvantages that the use of such educational software may have. We finally conclude with examples demonstrating the use of these techniques.

**MSC Subject Classification:** 00A35

Divided differences | 115$-$119 |

**Abstract**

This article deals with Newton divided difference interpolation polynomials. Textbooks in numerical mathematics where such polynomials are studied usually put the emphasis on numerical problems which are solved using these polynomials. Here we show that such polynomials can also be useful in solving some algebraic problems. In order to present this concept we shall first define the divided differences in the new sense: the divided difference of order $n$ is considered as a function of one variable and $n$ parameters. After the concise presentation of the theory of divided differences, we shall solve the problem of interpolation by integer polynomials. At the end we give the solution of an interesting problem using this theory.

**Keywords:** Divided differences, interpolation polynomial.

**MSC Subject Classification:** 00A35