In this paper, we present several different approaches to formula for the sum of integer powers of two in accordance with different representations of this sum or different algebraic methods for its computation. Our long-term experience shows the effectiveness of discussion on this theme for enhancing interest and creative thinking of the students about solutions of various problems, not only in mathematics but also in others fields of knowledge.
The absolute value operation is, by definition, very simple, but a composition of this operation with other functions, and even with a linear function, is difficult to the students. The reason is that this work involves many technical steps. In this paper we give a shortcut to graphical solutions of such problems. We also offer the absolute value as an attractive tool to obtain different symmetries on graphs of equations.
This article discusses the use of a scientific calculator in teaching calculus by using representations of mathematics notions in different sub-languages (analytical, graphical, symbolical, verbal, numerical and computer language). Our long-term experience shows that this may have a positive and significant effect on the enhancement of conceptual understanding of mathematical concepts and approaches. This transcends the basic computational uses, and implies a potential for real improvement in the learning success, cognitive motivation and problem solving skills of the student. We illustrate the steps we have taken towards doing this through some examples.
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