“A square is never a rectangle.”

The question below asked kids to state whether the given sentence is always, sometimes or never true. Then, they were asked to justify their response.


Suppose someone said this to you. How would you respond?

Many thanks to Tina for the submission!

7 replies on ““A square is never a rectangle.””

The student is right if they are talking about the common usage of the words outside of a mathematical discourse. If we see a square we don’t call it a rectangle.

So I would tell the student the way in which they are right, and the way in which they are wrong. To help them see that a square *can* be one kind of rectangle, compare to dogs and mammals. Someone asks what I see. I say I see a dog. I never would say that I see a mammal if I see a dog. But yes, it is a mammal. Similarly, your square is part of a larger group / classification, it’s also a rectangle. I’d work with them on the logic of subsets.

One of the other issues here, besides what Sue has brought up, is how students are generally taught classification of objects in early geometry. I have talked to my colleagues who teach early geometric classification and I emphasize that the purpose of this unit is for students to begin how to formally classify objects, and that the objective actually is NOT to learn individual definitions for each of the geometric shapes, in grade 3 or 4, the idea of classification is far more important than the specific definition of rectangle or square.

Unfortunately my suspicion is that most of the time people focus on teaching one definition of square, and another of rectangle, and do not work on the abstract idea of classification at all. The chief advantage of using classification over individual definitions is that you can see that a square and a rectangle actually share an enormous number of similar properties and are much more similar than they are different.

This is exactly the problem I’m facing: students have learned that a rectangle is two pairs of equal sides. That’s the definition and they’re sticking to it. The right angles are an after thought, and convincing them that a square does have two pairs of equal sides is not always effective (especially since we use the disjoint definitions of trapezoid and parallelogram!).

Totally agreed with Sue VH. This is a mistake I have made teaching for years: not recognizing and explicitly explaining that mathematical english logic and standard english logic are different. Standard english logic is pretty much self-consistent, just different (and not so convenient for math).

In standard english logic, “quadrilateral with two sides equal” means “quadrilateral with EXACTLY two sides equal” – this usage is consistent and sensible. Given four cats of the same color, if someone said “two of these cats are the same color”, in standard english logic everyone would agree the statement is false, because it means “two AND ONLY TWO of these cats are the same color”. But of course in math “quadrilateral with two sides equal” means “quadrilateral with AT LEAST two sides equal”.

The problem is that many of us teachers don’t make this explicit, because we are so used to it ourselves. We just say “quadrilateral with two sides equal” and expect students to catch on. Worse, I used to say that the standard english interpretation was wrong, which of course is completely confusing, it is a different convention but not at all wrong. Now that I have learned to be explicit about this, I rarely encounter this problem anymore.

Looking at this i would tend to assume that this child is of a low set. If a child came up with this i would celebrate what they have achieved. Depending on the set i would push them further. But again if it is a child of lower ability and they came up with this, i probably wouldn’t care if it was not explicitly dealing with sets and sub sets of quadrilaterals.

At a deeper level, I believe this boils down to exclusive vs inclusive definitions in mathematics, and unfortunately for our students, we teachers are not at all consistent in our usage. There are some people and textbooks who define rectangles (explicitly or implicitly) as right-angled quadrilaterals with opposite sides congruent that aren’t congruent to each other–from this perspective, squares are NOT rectangles.

On my point of exclusive vs inclusive definitions, consider Trapezoids. Some define trapezoids as quadrilaterals with EXACTLY one pair of parallel sides (a la Euclid, an exclusive definition) while others define trapezoids as quadrilaterals with AT LEAST one pair of parallel sides (an inclusive definition). I ‘blogged on this a year ago ( and got some pretty tough pushback from some readers who were perfectly happy to call a square a rectangle while denying vehemently that a parallelogram could ever be a trapezoid.

David Wees elegantly captured in his comment above the point I’m trying to make when he noted the difference between classifying and defining. I think David’s “classifications” may be what I’m calling “inclusive definitions”.

In the end, I suspect that we all need to be very careful with all of our language about classifying (or defining) all mathematical objects (geometric shapes, functions, properties, etc.). It is very easy for students not used to mathematical logic to get confused by our sometimes-inconsistent non-mathematical spoken language. As a side point, I hope for more frequent use of inclusive classifications because, as David notes, so many objects “actually share an enormous number of similar properties and are much more similar than they are different.” To be exclusive is to deny ourselves and our students many beautiful connections.

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