Posts Tagged ‘rectangle’

Is a Square a Rectangle?: Welsh edition

February 6, 2009

byrnes-euclid-rectangleAnswer:  No.

I’ve posted before about how even though in the US we define squares to be equilateral rectangles, there are many forces (often in the form of picture books on shapes) that treat squares and rectangles as distinct beings, and so it is really no surprise that many students reach college a little uncertain.  The mathematical definition doesn’t match the cultural one.

Anyway, this week in Geometry I was going over Euclid’s definitions, and I pointed out that he wrote explicitly that rectangles (oblongs) weren’t allowed to have four equal sides, which is different than the definitions we use today.

Then one of my students spoke up.  Katie is a senior math major getting certified to teach elementary & middle school, and this past fall she did part of her student teaching in Wales  (courtesy of a study abroad program here at the college).  She was in a 5th grade class, and according to their formal curriculum squares are not rectangles.  Indeed, Katie said that the definitions they used were pretty much the same as what appears in the translation of Euclid (rectangles have four right angles but can’t be equilateral; rhombi have  four equal sides but can’t have right angles; parallelograms have parallel sides, but can’t be equilateral or have right angles).

Similarly, when she taught about diamonds, she couldn’t call them squares even if they had four equal sides and four right angles.  She had to prove the same rules twice (say, that the diagonals of a square are equal, and that the diagonals of a equiangular diamond are equal) and when she drew them, she had to add a line to show whether the figure she was referring to was a square or a diamond.

square-diamond

That explains this Failblog post from last October:


more fail, owned and pwned pics and videos

(I’m not sure if this picture was from Wales, but she did see Shreddies in the store there.  She really liked the frosted kind.)

So now I’m wondering:  how are geometric figures (squares versus rectangles and the like) defined in other countries?

The image above is from Oliver Byrne’s way-cool color edition of Euclid’s Elements.

Is a square a rectangle?

June 6, 2008

I like this question. My first reaction — since I get this pretty much every semester that I teach a problem-solving or geometry class — is to ask what the definition of a rectangle is. Most people respond that it’s a quadrilateral with 4 right angles, maybe they add something about the opposite sides being parallel and/or equal, and then I ask if a square fits that definition. They answer yes, and the problem is solved.

But I think the question is really a little more subtle than that. In all the children’s books that we’ve acquired on shapes, none of them show a square on the rectangle page. Years of reinforcement that squares and rectangles are different shapes is hard to overcome with a single definition.

Furthermore, when I started teaching Geometry I learned that 2300 years ago Euclid didn’t define rectangles (which he called oblongs) in quite the same way as we do. Here’s a page from Oliver Byrne’s 1847 translation of Euclid’s Elements, which is one of my favorites because Byrne sure liked his color markers. He uses oblong the way we use rectangle.

Notice that Euclid said that an oblong did not have all four fides equal: a fquare was a completely different beast, not a special kind of rectangle. Euclid kept this distinction with all his geometric figures: a rhombus couldn’t have right angles (so a square wasn’t a special kind of rhombus either), a parallelogram (rhomboid) did not have right angles or equal sides, and an isosceles triangle had exactly two equal sides, not at least two. At Euclid’s Geometric Figures party when the figures divide into teams, the squares knew EXACTLY where to go, and it wasn’t with the rectangles: it was a partition, rather than a Venn diagram.

Another place where geometric problems can occur is with triangles. I think of the stereotypical triangle [in the US -- is it true in other countries as well?] as being one with a horizontal base, and probably isosceles.

But, just like the definition of rectangle, that hasn’t always been the case. In in “Words and Pictures: New Light on Plimpton 322″, Eleanor Robson explains, “if we look at triangles drawn on ancient cuneiform tablets like Plimpton 322, we see that they all point right and are much longer than they are tall: very like a cuneiform wedge in fact.”

Neither triangle is better or worse than the other, but they are different, illustrating the cultural influence on mental images of shapes. I find that intriguing.

I believe that the page of Byrne’s translation is fair to include because its over 70 years old. And an edition only sold for $300 in the ’70s — can you believe it? Not that I had more than $5 at any one time in that decade, but still, if I had and I wasn’t buying dollhouse furniture, I’m sure I would have bought it.


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