# Rhombuses, kites etc

As a high school teacher, I sometimes wonder about the usefulness of certain topics. Some topics seem to be in the textbook because they have always been there, not because they lead anywhere interesting.

For instance, I am fairly sure that rhombuses and kites are pretty useless. In fact, once you get past alt-int angles, parallelograms are not horribly useful later. I do not recall needing any of this in any math afterwards, at least up to and including calculus.

Since I have been told to cut out some geometry to make way for statistics / probability, it seems to me that rhombuses, kites and a good lot of parallelograms are perfect candidates for the chopping block.

Am I right? Or do rhombuses and kites turn out to be really useful in the conceivable future of any random student? I am not denying their beauty etc.

If proofs could be be put back into the state test, of course, rhombuses etc. would just be more practice in proofs.

• I see the use in helping students understand definitions and quantifiers; many high school students STILL struggle with the classification of quadrilaterals and recognizing that, for example, every square is a rectangle, but not every rectangle is a square. This facility with language (not just mathematical language) is very important for a logical thinker to have, and I have found that quadrilaterals lead to the best opportunity to develop it in geometry. – Opal E Jul 11 '19 at 19:30
• Students who will encounter vectors will also encounter parallelograms in one of the standard definitions of vector addition and in the connection between areas and 2-by-2 determinants. – Andreas Blass Jul 11 '19 at 20:13
• The same question on math.se math.stackexchange.com/questions/3290170/… – llllllllllllllllllllllllllllll Jul 11 '19 at 22:06
• – llllllllllllllllllllllllllllll Jul 11 '19 at 22:07
• @OpalE -- Your comment would make a good answer. – Jasper Jul 12 '19 at 22:00

Parallelograms are useful for understanding:

• Paths taken by light, especially through a layer of a medium with a different refractive coefficient
• Shear, and related deformations
• Area = height * width (but not necessarily the product of the sides' lengths)
• Dot products
• Surface integrals

Paths taken by light are useful for understanding which routes people and other animals will take to minimize their effort.

• I use parallelograms to explain "git rebase." – shoover Jul 12 '19 at 16:45

I am just now close to completing my first time teaching geometry (at a community college, a course that parallels the typical high school course) Using the fact that parallelograms (and hence rhombuses) have bisecting diagonals makes a short proof for why the line segment midpoint construction works. Other than that, I don't see rhombuses as a big part of geometry. I would keep some of parallelograms in. Have you looked at the common core standards to see which topics are meant to be kept?

Henri Picciotto, a former high-school teacher and (current) curriculum developer, has written a well-thought out pair of blog posts, In Defense of Geometry, which you might find useful.

• I wish I could upvote, but Henry mentioning NCTM positively and suggesting teaching fractals in public school (a topic that requires university-level education) did not jive with my outlook on school math education. – Rusty Core Jul 11 '19 at 21:09
• @RustyCore I'm not sure why you believe that "teaching fractals" (whatever that phrase means) requires a university-level education. Give me any topic in mathematics (with which I, personally, have sufficient knowledge), and I can prepare a lesson that I can teach to middle schoolers, and I can also prepare a lesson which might give some PhDs a hard time (elementary topics can become research level topics very quickly). I have a 50 minute "Introduction to Dimension Theory" talk that I have given successfully to high school freshman and sophomores in the past which touches on fractal geometry. – Xander Henderson Jul 11 '19 at 21:58
• @XanderHenderson He says that he does not don’t "the underlying mathematics" of statistics, yet he is going to teach high school students topology? Hausdorff dimension? Extended real numbers? I don't think so. It will be an overview with pretty pictures at best, all at the cost of removing either Algebra II or Geometry topics. This is not what public school students need. They need instead a coherent curriculum without repetitions and omissions, not some fancy stuff that school teachers know nothing about. – Rusty Core Jul 12 '19 at 2:46
• You can have your opinions, but his work is stellar, and helped me teach a good course. – Sue VanHattum Jul 12 '19 at 2:49

The following remarks do little more than amplify on a comment made above by Andreas Blass.

A basic use of parallelograms is to represent the sum of vectors in the plane as the diagonal of the parallelogram they determine.

Visualizing the action of a linear transformation of the plane can be achieved by examining the parallelogram determined by the images of the standard basis vectors. This is helpful for understanding the qualitatively different sorts of linear transformations.

Another use of parallelograms is when one formalizes notions such as area and volume, as is typically done in university courses in linear algebra and calculus. The best way to define the determinant of a $$2 \times 2$$ matrix is as the signed area of the parallelogram generated by the rows (or columns) of the matrix. The basic properties of the determinant - its antisymmetry and linearity in the rows/columns of the matrix and its behavior with respect to elementary row operations can be interpreted in terms of orientations of parallelograms and cutting and pasting parallelograms and their subtriangles. All of this is useful for students who will study things such as the theorems of Green/Stokes, surface integrals, and so forth, and need to assimilate notions such as orientation and signed area. For example, motivating the definition of the integral of a function over a surface by constructing a Riemann sum approximating a parametric form of the integral requires interpreting the summands of the Riemann sum as areas of infinitesimal parallelograms in planes tangent to the surface.

I would keep the parallelograms. Too much application in SAT questions--they love the vertical angle theorem.

Agreed on ditching kites and rhombuses. (In the interest of time.) I have never used them in engineering, physics, other math classes that support engineering/science. Actually, I don't even remember covering them much in 1980s geometry. If we did, it was super quick (a day at most).

You could also probably cut solid geometry some (even though this pains me to see this--just give them a handout with the mensuration formulas and don't even test it).

Really, even proofs could be cut. (I'm not saying to cut it all...but if the point is exposure to proofs, do you really need so much for "exposure".) Geometry is such a diversion from the other courses in their headlong build towards calculus.

Anyone saying to keep the topics, ought to suggest other places to cut. Hours are zero sum. And not doing the stats doesn't sound like an option. But it would help if you told us how many days you are trying to free up (how long spending on stats).

• The question for me is, how much time is saved by skipping these couple figures? I graduated HS in ‘80, and don’t recall “kites”, but seeing they were in the curriculum in ‘13 didn’t bother me. ‘Rhombus’? A parallelogram with equal sides. 2 extra seconds to the lecture. (So, +1 for well articulated answer that I sort of agree with) – JTP - Apologise to Monica Jul 15 '19 at 0:16
• @JoeTaxpayer More than the definition is taught. Properties of a rhombus are also taught, e.g. the diagonals of a rhombus bisect each other at right angles, to find the area multiply the lengths of a diagonal and divide by 2, etc. There are many problems with these properties. – Amy B Jul 25 '19 at 5:43