Kawasaki (2001) used the ‘Problem e’ below, to clarify the nature of geometrical sense.
Problem e : Find α + β in the figure in which three congruent squares are placed side by side.
Independently, we found the formula Tan −1 1/2+Tan −1 1/3=π/4 (Euler’s formula) in Izumi (1961), and we recognized that the problem e is a geometrical version of Euler’s formula. We felt that the problem e which is related to Euler’s formula has interesting geometrical aspects and the value in mathematics education. In this article, we consider the nature of the problem e and its usefulness and possibilities as teaching material in junior and senior high school mathematics classes.
In Section 2, we considered a variety of solutions to the problem e and classified them by learning contents. As a result, we came to the conclusion that the problem e can be used variously both in junior and senior high school mathematics classes and will be a valuable teaching material.
In solving the problem e geometrically,“the viewpoints of the sum of the measure of angles”are the key to the success. We considered “the sum of angles” from three viewpoints :
Viewpoint 1 : Adjoining two angles. This is the most natural way. The examples of the solutions to the problem e from this viewpoint are shown in 2.1.1.
Viewpoint 2 : Reducing to the fact “the measure of an external angle is equal to the sum of the measures of two inner angles which aren’t adjacent the external angle”. We recognized that this viewpoint is very useful. In fact, this viewpoint is a fundamental idea in this article. The examples of the solutions to the problem e from this viewpoint are shown in 2.1.2 and 2.2.3. We discussed in detail the geometrical proofs of Clausen’s formula which is more complicated than Euler’s formula in 3.2 of Section 3. Viewpoint 3 : Direct and effective way which doesn’t use the movement of angles to consider “the sum of the measure of angles”. In viewpoint 1 and 2, we need to “move the angle adequately”. We found the way from
viewpoint 3 which dosen’t need to move the angle by developing the way in 2.2.3. We formulated this way as the lemma in 3.3 and we showed the proofs of the formulae which are more complicated than Euler’s formula.
In Section 4, we developed some problems which are related to the problem e from the viewpoints 2 and 3. As a result of the consideration in this article, the viewpoints and ways concerning the problem e will be a valuable teaching material that makes junior and senior high school students enhance their interest in geometry.
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