This lesson has the potential to address the mathematics objectives stated above. We provide lesson-implementation details, sample student work, and student response to the lesson.

In this article, we describe an impossible-geometry lesson where two high school classes in different Northern Nevada school districts (an integrated algebra/geometry/statistics class comprised mainly of freshmen and a tenth-grade geometry class) explore the concept of impossible geometry before creating their own impossible object. In Victoria, Shape, Space and Design is a Foundation Mathematics area of study that includes investigation of the properties of two- and three-dimensional shapes and two-dimensional representations of three-dimensional objects (Victorian Curriculum and Assessment Authority, 2005). The Linking Two and Three Dimensions component of Queensland's Mathematics A Senior Syllabus (Queensland Board of Senior Secondary School Studies, 2000) addresses the importance of practical knowledge of various construction areas and asserts, 'Particular emphasis should be given to the representation of three-dimensional constructions in two dimensions' (p.

Spatial explorations involving two- and three-dimensional shapes are an important part of school curricula. Swedish artist Oscar Reutersvard, however, is considered to be the 'the father of impossible objects' due to his deliberate, prolific work in this area. Necker discussed impossible objects in an 1832 journal article, and Penrose and Penrose particularly popularised the idea in their 1958 publication on the topic. Impossible geometry has been a subject of interest for some time. Whereas individual parts of the figure may be possible, the object in its totality is not. It is thus an optical illusion where the misperception requires careful analysis to detect and explain. One innovative area of recreational geometry that has rich potential to engage and challenge a wide variety of students is 'impossible geometry.' An impossible geometric object is a two-dimensional drawing that gives the impression of a viable three-dimensional figure, which in fact cannot exist. Australian students thus need stronger preparation in geometry. Scores for Year 8 students fell significantly below the TIMSS scale average. Geometry is an area in which Australian students performed particularly poorly on the 2007 Trends in International Mathematics and Science Study (TIMSS) (Thomson, Wernert, Underwood, & Nicholas, 2008).

Impossible world impossible shapes