By Sue Johnston-Wilder, John Mason
This article and interactive CD-ROM aid academics expand their educational practices via leading edge methods for educating geometry as constructed via the Open University's Centre for arithmetic schooling.
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Extra info for Developing Thinking in Geometry
It is not always possible to draw an interior rectangle if you start from any point on the longer side. Since it is possible to construct an interior rectangle from any point on the shorter side of the original rectangle, this process can be repeated, as in the next task. 3 Sequence of Rectangles As before, imagine drawing a rectangle inside a given rectangle. Now imagine drawing another rectangle inside this one and continuing the process. What conclusion can you draw about this sequence of rectangles?
Sometimes it is necessary to insert or construct extra elements that enable you to discern sub-figures whose known relationships (for example, similar triangles and ratios and parallelism) you can use to make deductions about other relationships. Notice also that it is important at the end of a chain of reasoning to look back at what was invariant, and to clarify what it is that is allowed to change and in what ways. 4 Look back over the reasoning given in this section and consider what particular features of language are involved in reasoning on the basis of properties.
If a second circle is added, as suggested in the comment above, then you could see the diagram as consisting of two similar figures. Each figure is a square with a circle inside it. Since the ratio of the areas for the two squares has already been established, it follows that the ratio for the two circles will be the same. In his book How to Solve It, George Polya (1957) poses the problem illustrated in the next task. 3 Polya’s Square–Triangle Problem Given a triangle, draw a square inside the triangle such that two of its vertices are on the base of the triangle and the other two vertices are on the other two sides of the triangle.