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As a teacher, I remember endlessly talking about squares in the same breath as saying “squares are a special type of rectangle”. I often designed activities that resulted in pupils producing tree or Venn diagrams to show the family classifications of quadrilaterals. Whether it was angles, dimensions, measures, constructions, or to represent unknown quantities, shapes could regularly be seen in lessons.
As I design the geometry waypoints in the Cambridge Mathematics Framework I have found a large amount of research concerning the classification of quadrilaterals. 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. Pupils rarely fully understand or know a true mathematical definition for each quadrilateral, and instead tend to list their characteristics, four sides and all.
A few specifics stick out. Some muddy the waters while others help us to clear up the problems. With that in mind, here’s a small selection of important issues to think about when you’re working in this area.
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. 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.
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!
Euclid, 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. All other quadrilaterals were trapezia.
Even 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. Each is either a square, an oblong, a rhombus, a rhomboid or trapezia. Now wouldn’t that make life simpler?
Well yes and no. 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?
It all comes down to what we can deduce and infer from one shape to another. A square is a special type of rectangle and rhombus and therefore a special 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).
In other words: a theorem you prove for a parallelogram holds for squares, rectangles, and rhombi as they are all types of parallelogram. Yet a theorem that holds for a square may not hold for all parallelograms as not all parallelograms are squares.
At 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. It may be that your proof based on the square doesn’t rely on these tightened restrictions that make it square not just parallelogram, hence actually your proof will work for all parallelograms.
When searching through the wonders of the web, Wikipedia offers this wonderful diagram:
You may not completely agree with the names being used, but what’s wonderful is how you can identify the tightening or loosening restriction as you take a walk around. Leave a region and you loosen, enter another layer and you tighten. It also highlights that actually, you know what, maybe oblong isn’t such a nasty word – oblongs and squares make up the rectangle family and ‘oblong’ might help with the whole squares are rectangles confusion. Alternatively, Clements and Sarama (2009) suggest using the double name square-rectangle. Would this mean we would also have rhombus-parallelograms? It would be nice to consider if some regions are actually empty and as a team we have a question as to whether ‘darts’ here should be encompassed in the kite region too?
The whole conversation just highlights how confusing defining quadrilaterals can be. Crucially we need to decide on what we consider to be necessary and sufficient conditions, and therefore familiar relationships, whilst at the same time being ready to state them explicitly. Maybe once we’ve done that we can draw our own Wikipedia diagram for our definitions. I’ll leave that once with you, but I would be interested in what you design! Can I uncover your definitions from your diagram?
Clements, D.H., Sarama, J., 2000. Young Children’s Ideas About Geometric Shapes. Teaching Children Mathematics 6, 482–488.
De Villiers, M., 1994. The role and function of a hierarchical classification of quadrilaterals. For the learning of mathematics 14, 11–18.
Sarama, J., Clements, D.H., 2009. Shape, in: Early Childhood Mathematics Education and Research, Learning Trajectories for Young Children. Routledge, New York, pp. 199–246.
SOMETHING TO TRY:
KS1: What does each set of shapes have in common? What makes each one different?
KS2: A rectangle is a special type of parallelogram. Why?
KS3: Draw a tree diagram to link the family of quadrilaterals. Explain the links you have made.
KS4: Construct a cyclic parallelogram.
KS5: Van Aubel's theorem states that: If squares are constructed on the sides of any quadrilateral, then the line segments connecting the centres of opposite squares are equal and perpendicular. What shape would these centres form if the original quadrilateral was a parallelogram? Using hierarchical classifications show why. Consider the proofs of the original theorem here.