This is for junior-level (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.

This issue is, at least in Math, the choice between

1. exclusive definition: A rhombus has four equal sides but NO right angle; i.e. the class of rhombi EXCLUDES the class of square;
   
2. 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.

1. Inclusive definitions are, in general, the way that mathematicians work, and the way that students will experience definitions later in mathematics;
   
   
2. 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.
   
   
3. 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'!
   
   
4. 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.
   
   
5. 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.

1. Both definitions agree you want one pair of parallel sides and a second pair of sides of equal length.
   
   
2. You do not want a typical parallelogram in your set.
   
   
3. EXCLUSIVE definition: Only one pair of parallel sides. INCLUSIVE definition: There is a mirror joining the mid-points of the two parallel sides.
   
   
4. 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 half-turn 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.

1. They are easier to work with for actual reasoning.
   
   
2. They better match certain kinds of construction (e.g. origami).
   
   
3. Working with symmetry as the core of geometry is the 'modern' (since 1870) way in which mathematicians work.

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

Go to Math Central