This is for juniorlevel (grade 6) mathematics.
Can a square be a rhombus? Some sources say yes, some
say no. Some sources define a rhombus as a quadrilateral and parallelogram
with equal sides, but without right angles. Some sources say a square
is a special case of a rhombus. Clarity, please!
Beth (teacher) is asking
Thanks.


Hi Beth,
This issue is, at least in Math, the choice between
 exclusive definition:
A rhombus has four equal sides but NO right angle; i.e. the class of
rhombi EXCLUDES the class of square;
 inclusive definition: A
rhombus has four equal sides; i.e. the class of rhombi INCLUDES all
squares.
There are three points I would make for WHY the inclusive
definition is what should be used in mathematics classrooms.
 Inclusive definitions are, in general, the way that
mathematicians work, and the way that students will experience definitions
later in mathematics;
 When you look at how people reason, even how textbooks
which start with exclusive definitions reason, you find that our reasoning
uses inclusive definitions. For example, if you used either definition
of a rhombus (the four equal sides property) and proved that opposite
angles are equal, you would NOT ever use the property that the angles
are not right angles. The PROOF applies to the inclusive class of objects.
The RESULT applies to squares as well.
 If you use some dynamic geometry software, such as
Geometer's SketchPad, and construct a rhombus, you will find that as
you drag your points around, maintaining the equality of the sides,
that a square will appear as one of the objects. You do not want to
say that the construction is now 'wrong'!
 Use origami to 'fold out a rhombus', by taking one
central fold, then another fold at right angles. Now you have a central
corner where both folds meet, and have four layers of paper. Fold over
that corner by any amount to create a 'triangle'. Unfold it all. You
will find the the last fold (of four layers) now forms a nice rhombus,
with the two original folds as 'mirrors' of the object. (These are the
mirrors that show the opposite angles are equal.) This is an example
of a 'symmetry definition' of a rhombus  something with two mirrors,
on lines joining opposite vertices. Again, this is an inclusive definition.
 Here, inclusive definitions better match the way we
think visually (and kinesthetically or hands on). The exclusive definition
better matches how we think verbally or linguistically. It turns out,
when you use brain images to watch people doing many basic mathematical
tasks, important parts of mathematical reasoning are done in the parts
of the brain associated with visual / kinesthetic reasoning.
A key example for inclusive and exclusive definitions
for types of quadrilaterals is the problem of an isosceles trapezoid.
 Both definitions agree you want one pair of parallel
sides and a second pair of sides of equal length.
 You do not want a typical parallelogram in your set.
 EXCLUSIVE definition: Only one pair of parallel sides.
INCLUSIVE definition: There is a mirror joining the midpoints of the
two parallel sides.
 The exclusive definition says that a rectangle is
not an isosceles trapezoid. The inclusive definitions says that it is!
Here, because people are NOT used to using symmetries
such as mirrors, rotations (e.g. parallelogram as a quadrilateral with
a halfturn rotation) etc. as defining properties of quadrilaterals, they
are usually trapped into using an exclusive definition.
However, there are three things to say about the symmetry
definiitions.
 They are easier to work with for actual reasoning.
 They better match certain kinds of construction (e.g.
origami).
 Working with symmetry as the core of geometry is the
'modern' (since 1870) way in which mathematicians work.
So if a student is later going to do geometry at a high
level, or need to use it in many applied areas, learning the symmetry ideas
(and the associated inclusive definitions) is a good basis for later learning.
The fact that different books give different definitions
can be used as a teaching opportunity. Let the students try both of them.
Let them do some of the exercises I suggested above  with origami, with
simple reasoning, with a computer program (if they can access it). Let
them debate. This is a way of learning that the way we talk about mathematics
is a human thing  developed over time.
If you want to stretch their minds (and perhaps yours!),
you could even play with the concepts on a sphere. The mirror and symmetry
definitions go right on through. All 'squares' (four equal sides and four
equal angles) do NOT have right angles. Use old tennis balls and elastics
where any elastic on an 'equator' (great circle cutting the sphere exactly
in half) is a straight line.
Walter Whiteley
York University

