Section 8-2: Tiling the Plane (Day 123)

Last night I tutored my geometry student. He is currently in Section 6-7 of the Glencoe text -- which came as a surprise to me, because I mistakenly thought that Chapter 6 had only six sections. And this chapter is on coordinate proofs.

This section corresponds to Section 11-1 of the U of Chicago text -- which distinguishes it from the rest of Glencoe's Chapter 6, as this corresponds to U of Chicago's Chapter 5. We can see the reason why Glencoe would include coordinate proofs with quadrilaterals -- many coordinate proofs, after all, are about quadrilaterals.

Notice that when I covered coordinate geometry back in January, I spent so much time trying to develop the distance, midpoint, and slope formulas that I ended up not giving sufficiently many actual coordinate proofs as a result. Of course, part of this is the rush that we're in to cover all of these chapters before PARCC. Next year, when I plan on fixing some of the mistakes I made this year, I will make sure that Section 11-1 is included as a separate lesson -- even if it has to occur after the rest of Chapter 11 (as, come to think of it, it logically should).

Because of this, I showed my student the Review for Chapter 11 worksheet. Actually, there are some good coordinate proofs on that review worksheet, and he was able to understand most of these after I explained what he was to do. But, once again, if this were an actual geometry course that I was teaching, of course I'd want coordinate proofs to appear in a separate Lesson 11-1 rather than appear suddenly on the Chapter 11 Test.

Well, this is why I give these worksheets to the student I'm currently tutoring. Only by giving my worksheets to actual students can their flaws be revealed to me.

Let's get back to the U of Chicago text. Section 8-2 of the U of Chicago text is on tiling the plane. A well-known geometric name for a tiling of the plane is a tessellation. Here's is how the U of Chicago text defines this term:

Definition:

A covering of a plane with congruent copies of the same region, with no holes and no overlaps, is called a tessellation.

Here's a song from that old PBS show Square One TV about tessellations:

Notice that the word tessellation doesn't actually appear in the Common Core Standards. But here are some standards that are often used to justify teaching tessellations in math classes:

CCSS.MATH.CONTENT.4.OA.C.5
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

[boldface mine, italics the author's]

CCSS.MATH.CONTENT.3.MD.D.8
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

You may be wondering what this standard has to do with tessellations. Well, it refers to the New York State curriculum -- I've referred to NY State in this past as being one that has developed a full Common Core curriculum for all grades (Engage NY). In particular, there is an activity where students tessellate regions in order to distinguish the boundary of the region -- which is where the perimeter is -- from the interior.

In some ways, the U of Chicago text is doing the same thing as the State of New York -- we see how Section 8-2, on tessellations, occurs right after Section 8-1, on perimeter. But there's one problem with tying this to the above listed standard -- notice that it's a third grade standard! And the other standard isn't much better -- it's a fourth grade standard. So I can see how including this lesson on tessellations in a high school geometry course based on Grade 3-4 standards would be a big problem!

So there are several issues here. First of all, a few months ago, I read another Common Core horror story where a parent whose third grader was learning tessellations instead of, most likely, the multiplication tables. And as so often happens, I read an article months ago when I'm nowhere near the relevant section of the text, and now that I've reached it and want to discuss it here on the blog, I can't find it no matter how hard I search on Google. (I do remember a comment that tessellations should appear in art class, not math class!) Instead, I found comments from a poster whose home state, Louisiana, formerly had state standards that required tessellations to be taught, and he was actually happy that Common Core would eliminate that wasteful Louisiana standard!

Anyway, even though I can't find the website, let me comment on this issue. First of all, let's recall that Grade 3-4 is right on our traditionalist-progressive boundary. On the traditionalist side of the boundary, in third grade, the focus should be on traditionalist math -- and this, above all, means the multiplication tables. Any third grade standard that takes away from memorizing the times tables should immediately be abolished.

But should tessellations be taught in high school geometry -- which is well on the progressive (or at least mixed philosophy) side of the boundary? One could argue that tessellations are, even in high school, a topic that takes time away from more important topics. Indeed, notice how I've juxtaposed today's lesson on tessellations -- viewed by many as a time-waster -- with a complaint that there wasn't enough time to teach coordinate proofs fully! You might say that if I really wanted to put my money where my mouth is, I should drop today's tessellation lesson and teach coordinate proofs instead! But I don't believe that a coordinate proof lesson belongs today, between the perimeter and area lessons.

Well, part of this is due to timing. Recall that my insistence that most of my tests fall on Fridays goes back to the students' inability to study for a hard test over the weekend. (But don't forget that our next test will be a week from Thursday, not Friday.) Forcing all tests to fall on Fridays means that we run out of time during weeks that contain a holiday. The week that we covered coordinate geometry contained the MLK holiday, while this week's lesson on tessellations is during the Long March when there are no school holidays. And so I end up skipping or squeezing lessons during the holidays and including extra lesson during five-day weeks.

But I also believe that including tessellations with Common Core lessons make sense. As we know, there are four types of transformations: reflections, rotations, translations, and glide reflections. We know that there exist figures that are reflection-symmetric (like isosceles triangles and trapezoids) and rotation-symmetric (like parallelograms). But then this leads us to wonder, do there exist figures that are translation-symmetric or glide-reflection-symmetric?

As it turns out, tessellations are translation-symmetric. Indeed, one can see that while reflection- or rotation-symmetric figure can be finite, a translation-symmetric figure must be infinite. To see why, let's consider when a figure is symmetric with respect to some transformation:

For F a figure and T a transformation, F being T-symmetric means that for every point P, F contains the point P if and only if it contains the point P' = T(P).

Notice that applying above definition repeatedly implies that if a T-symmetric figure contains a point P, not only must it contain P' = T(P), but also P" = T(T(P)), P'" = T(T(T(P))), and so on. Now if T is a reflection, then T(T(P)) is just P by the Flip-Flop Theorem, and so there are really only two points, P and P'. If T is a rotation, T iterated a small finite number of times may also be the identity, and even if it isn't, at least all the points P, P', P", P'", lie on a circle whose center is the same as that of the rotation T.

But if T is a translation, then P, P', P", P'", ..., are not only all distinct, but are linearly increasing in distance from the original point P. This is obvious if we view translations as vectors -- if v is the translation vector of T, then T iterated n times is a translation whose vector is nv -- and as the scalar n increases without bound, the magnitude of the vector nv increases without bound as well. Thus if a translation-symmetric figure contains at least one point, it must contain infinitely many points. And if a translation-symmetric figure contains at least one triangle, then it must contain infinitely many triangles -- often making it a tessellation of triangles. QED

And so we include tessellations for completeness -- since without them, we don't have figures that are translation-symmetric just as we have reflection- and rotation-symmetric figures. For completion, there also exist glide-reflection-symmetric figures. Since the composite of a glide reflection with itself is a translation, GR-symmetric figures may also be tessellations, but not all tessellations are GR-symmetric. In the U of Chicago text, the tessellations that include kites tend to be GR-symmetric. (Think about it -- the kites are already reflection-symmetric, and they are translated to form the tessellation. The reflection and the translation gives us a glide reflection.) The important thing to see here is that tessellations are mathematically rigorous objects -- not just designs to draw in elementary art classes.