Find the area of ABCD.
(Here is some given info from the diagram: Angle A = 30, angles C, D are right angles. Also, there are some additional points that aren't labeled, so let me label them myself. BE = EA = 3, CF = FD, G is the foot of the perpendicular from E to
First, we notice that AEG is a 30-60-90 triangle with hypotenuse 3, and so EG, the side opposite the 30-degree angle A, must be 3/2.
Now imagine cutting off triangle AEG and rotating it 180 degrees about E. Then A' is exactly B, and G' is the vertex of CDGG', a rectangle with the same area as ABCD. Then G'G, its height, is exactly twice as much as EG = 3/2, and so G'G = 3. Its base is DG = 6. Thus its area is 18.
Therefore the area of ABCD is also 18 square units -- and of course, today's date is the eighteenth.
We are currently in Chapter 8 of the U of Chicago text, and except for the part where a 30-60-90 triangle was needed to find EG, this is a Chapter 8 problem. In fact, notice that Quadrilateral ABCD is a trapezoid, and so this suggests the use of the trapezoid area formula of Lesson 8-6.
But notice that I didn't actually use the trapezoid area formula in this problem. Instead, I cut off a small triangle and rotated it to form a rectangle. This seems like something we'd do with students to introduce the trapezoid formula, or to get them to find the area of the quadrilateral without teaching them a trapezoid area formula. We're adults here, so shouldn't we just use the formula?
OK, then, try using the formula A = (1/2)h(b_1 + b_2). You'll find that the bases of the trapezoid are b_1 = BC = 6 - 3sqrt(3)/2 and b_2 = AD = 6 + 3sqrt(3)/2. And now you see why I avoided using the trapezoid area formula.
Some texts, though not the U of Chicago text, point out that the area of a trapezoid is the product of the height and the midsegment of the trapezoid. (Our text only refers to the Midpoint Connector Theorem for triangles in Lesson 11-5. Midpoint connectors for trapezoids aren't mentioned, and the term "midsegment" isn't defined for triangles or trapezoids.)
In this problem, both bases are irrational numbers requiring radicals to write exactly, while the midsegment is a simple whole number. This is the only type of trapezoid area problem where it's much better to avoid the classical area formula in terms of b_1 and b_2. We should use either the method I used above, or introduce midsegments.
If you prefer midsegments, then emphasize rectangle DFEG. We use 30-60-90 to find EG = 3/2, and then as opposite sides of a rectangle are congruent, EF = 6 and FD = 3/2. EF is already the desired midsegment, and FD is half of the height CD, and so CD = 3. Thus the area is 18 square units.
This trapezoid area problem definitely gives us something to thing about after winter break, when I must decide how to teach Lesson 8-6 after skipping Lesson 8-5. But let's get to today's lesson.
This is what I wrote three years ago about today's lesson:
Lesson 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.
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.
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 Lesson 8-2, on tessellations, occurs right after Lesson 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, I once read another Common Core horror story where a parent whose third grader was learning tessellations instead of, most likely, the multiplication tables. I have located this post on the Internet. I should have known all along that the original posters were the usual suspects -- Drs. Katharine Beals and Barry Garelick:
[2018 update: Back in March 2016, I added the "traditionalists" label to this post. I don't want today's post to be yet another traditionalists' debate post, so I'll cut out much of this. Let's just skip to "Niels Henrik Abel," since this leads to an interesting discussion.]
I've mentioned the poster "Niels Henrik Abel" in previous posts -- here's what he writes:
Niels Henrik Abel said...
I really don't understand the fascination with tessellations. Has manipulations with tessellations become a 21st century job skill?
Let's sum up all of these comments by asking the question -- should tessellations be taught at school, and if so, when?
So let's return to the question -- when should tessellations be taught? I've already given the rationale for teaching tessellations in high school Geometry -- to demonstrate translation symmetry. So despite the poster "lgm" saying that most people have already figured out tiling in kindergarten, is it really true that kindergartners understand the symmetries of the Euclidean plane?
Most traditionalists, of course, don't believe that plane transformations should be taught at all -- instead the properties of parallel lines would be taught as postulates. But given that transformations must occur in high school Geometry, how much of it should be taught in the lower grades? After all, many Geometry topics such as perimeter and area don't appear for the first time in high school Geometry -- instead they appear throughout the curriculum of Grades 3-8.
We know that transformations themselves first appear in eighth grade math. But I can easily understand the idea of teaching tessellations in upper elementary school -- it is one of the easiest parts of transformation geometry for such a young student to understand,
Let's look at what "Abel" wrote again:
Niels Henrik Abel said...
I really don't understand the fascination with tessellations. Has manipulations with tessellations become a 21st century job skill?
No, tessellations are not a modern job skill. Here "Abel" is suggesting that, given the Common Core's stated goal of preparing students for college and careers, tessellations should not be taught. In particular, in Grades 3-5, the only math that should be taught, that is directly related to college and career skills, is the study of the semiring of whole numbers -- that is, arithmetic. Any math covered in elementary school, especially the third grade and below, other than arithmetic is considered a waste of time. The geometry that appears in the third grade about perimeter (and sometimes area) is directly related to arithmetic (as in adding or multiplying lengths and widths). And as I say so often, I agree with traditionalists for most elementary math. I'd argue that tessellations, as translation-symmetric objects, can wait until eighth grade -- since, after all, that is the year when translations first appear in the curriculum.
Other than that, many things appear in the Geometry class that aren't directly related to jobs -- congruence, various proofs, and so on, yet traditionalists have no problem with those.
Let's conclude with Barry Garelick's post from that thread:
Barry Garelick said...
I had tessellations in my math book in 7th and 8th grades (in the 60's) but it was devoted to one page and emphasized that in a tesselation, the sum of the angles at a particular point is 360 degrees. Thus, certain shapes like squares and hexagons tesselate while octagons do not. Octagons and squares can be made to tessellate, however. This is probably the emphasis in Singapore's program,though I haven't seen it.
What others are talking about here are the tessellations that involve the nesting/interlocking of designs. Mathematics takes a back seat to the art aspect of the activity.
By the way, today's worksheet from three years ago contains four review questions. Two come from yesterday's perimeter lesson, but the other two review material from Chapters 5 to 7 (#8 on the angle of a regular pentagon and #10 on a triangle congruence proof). So at least the students will get two more review questions for the upcoming final.
Let's sum up all of these comments by asking the question -- should tessellations be taught at school, and if so, when?
- High School Geometry
- Grade 3-5 Art
- Grade 3-5 Math
- Kindergarten
- In grades K-4, math is a "story of units." That is, it's all about the semiring W of whole numbers.
- In grades 5-7. math is a "story of ratios." That is, it's all about the field Q of rational numbers.
- In grade 8, math expands to describe the field R of real numbers.
- In high school Algebra, math is a "story of functions" It describes the field C of complex numbers, the ring R[x] of polynomial functions, and its quotient field of rational functions.
- In high school Geometry, math describes the symmetry group of the plane. This group is often known as the Euclidean group of the plane, because it distinguishes Euclidean geometry from the other types of geometry. Yes -- it's the existence and uniqueness of parallels that distinguish Euclidean from non-Euclidean geometry, but as we've seen here on the blog, all of the properties of parallels can ultimately be proved from transformations.
- reflections
- rotations
- translations
- reflection symmetry
- rotation symmetry
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.
So let's return to the question -- when should tessellations be taught? I've already given the rationale for teaching tessellations in high school Geometry -- to demonstrate translation symmetry. So despite the poster "lgm" saying that most people have already figured out tiling in kindergarten, is it really true that kindergartners understand the symmetries of the Euclidean plane?
Most traditionalists, of course, don't believe that plane transformations should be taught at all -- instead the properties of parallel lines would be taught as postulates. But given that transformations must occur in high school Geometry, how much of it should be taught in the lower grades? After all, many Geometry topics such as perimeter and area don't appear for the first time in high school Geometry -- instead they appear throughout the curriculum of Grades 3-8.
We know that transformations themselves first appear in eighth grade math. But I can easily understand the idea of teaching tessellations in upper elementary school -- it is one of the easiest parts of transformation geometry for such a young student to understand,
Let's look at what "Abel" wrote again:
No, tessellations are not a modern job skill. Here "Abel" is suggesting that, given the Common Core's stated goal of preparing students for college and careers, tessellations should not be taught. In particular, in Grades 3-5, the only math that should be taught, that is directly related to college and career skills, is the study of the semiring of whole numbers -- that is, arithmetic. Any math covered in elementary school, especially the third grade and below, other than arithmetic is considered a waste of time. The geometry that appears in the third grade about perimeter (and sometimes area) is directly related to arithmetic (as in adding or multiplying lengths and widths). And as I say so often, I agree with traditionalists for most elementary math. I'd argue that tessellations, as translation-symmetric objects, can wait until eighth grade -- since, after all, that is the year when translations first appear in the curriculum.
Other than that, many things appear in the Geometry class that aren't directly related to jobs -- congruence, various proofs, and so on, yet traditionalists have no problem with those.
Let's conclude with Barry Garelick's post from that thread:
What others are talking about here are the tessellations that involve the nesting/interlocking of designs. Mathematics takes a back seat to the art aspect of the activity.
By the way, today's worksheet from three years ago contains four review questions. Two come from yesterday's perimeter lesson, but the other two review material from Chapters 5 to 7 (#8 on the angle of a regular pentagon and #10 on a triangle congruence proof). So at least the students will get two more review questions for the upcoming final.
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