Yet a theorem that holds for a square may not hold for all parallelograms as not all parallelograms are squares.Īt this point it’s really interesting to look at the restrictions needed as you pass around the family of parallelograms to consider what remains invariant and how this interacts with the theorem in question. In other words: a theorem you prove for a parallelogram holds for squares, rectangles, and rhombi as they are all types of parallelogram. These hierarchical definitions lead to more economical definitions of concepts and formulation of theorems, simplify the deductive systematization and derivation of the properties of more special concepts, provide useful conceptual schema during problems solving, can suggest alternative definitions and new propositions and provide useful global perspectives (De Villiers, 1994). A square is a special type of rectangle and rhombus and therefore a special parallelogram. It all comes down to what we can deduce and infer from one shape to another. A question to ask yourself is why do we have the inclusive definitions we do? What’s the point – surely they just daze and confuse? Each is either a square, an oblong, a rhombus, a rhomboid or trapezia. All other quadrilaterals were trapezia.Įven just making sense of this is a great opportunity to really think about what these shapes look like and their familiar relationships as Euclid implies that there are in fact no intersections at all between shapes. Now I’m not in a position to make a definitive decision on this – but pointing out that these issues exist (especially in the multicultural classrooms in which we teach) is essential, as is pointing out to those who search the internet when lesson planning that some care, attention and scepticism is often needed!Įuclid, the forefather to much of our school geometry curriculum, defined (Book 1, Definition 2) a square to have equal sides and right angles, an oblong to have four right angles but not four equal sides, a rhombus to have four equal sides but no right angles, a rhomboid to have equal opposite sides and equal opposite angles but without right angles and without four equal sides. In the US (for some) a trapezium is a four sided polygon with no parallel sides in the UK a trapezium is a four sided polygon with exactly one pair of parallel sides whereas in Canada a trapezoid has an inclusive definition in that it’s a four sided-polygon with at least one pair of parallel sides - hence parallelograms are special trapezoids. What is the definition of a trapezium? Is it a shape with exactly one pair of parallel sides or at least one pair of parallel sides? Or maybe even none at all! Different cultures define a trapezium slightly differently and many have the term trapezoid too. With that in mind, here’s a small selection of important issues to think about when you’re working in this area. Some muddy the waters while others help us to clear up the problems. Pupils rarely fully understand or know a true mathematical definition for each quadrilateral, and instead tend to list their characteristics, four sides and all.Ī few specifics stick out. Much of this research identifies issues that we’re all too familiar with: pupils not recognising that a square is a type of rectangle and a rectangle a type of parallelogram the necessary and sufficient properties a quadrilateral and the characteristics of the shapes to list a few. Whether it was angles, dimensions, measures, constructions, or to represent unknown quantities, shapes could regularly be seen in lessons.Īs I design the geometry waypoints in the Cambridge Mathematics Framework I have found a large amount of research concerning the classification of quadrilaterals. I often designed activities that resulted in pupils producing tree or Venn diagrams to show the family classifications of quadrilaterals. As a teacher, I remember endlessly talking about squares in the same breath as saying “squares are a special type of rectangle”.
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