I showed my student my worksheet from Section 8-2 of the U of Chicago text. I mentioned earlier how the U of Chicago places tessellations in the perimeter and area chapter while Glencoe's placement seems a bit more logical. Tessellations have translational symmetry just as kites have reflectional symmetry and parallelograms have rotational symmetry. Indeed, Mario Livio mentions tessellations as translation-symmetric figures in the first chapter of The Equation That Couldn't Be Solved.
Oops! I said that I wasn't going to blog about Livio's book anymore. But I couldn't help but mention his book yet again since Chapter 1 of his book ties in so well to my student's lesson last night. The U of Chicago mentions that "tessellate" is Latin for small stone and that some of the first tessellations were the small stone floors of Roman buildings. Well, Livio dates translational-symmetric figures back to 17,000 BC, when a prehistoric Ukrainian bracelet had a repeating zigzag pattern. Notice that Livio never actually uses the word "tessellation," but it's clear that he's referring to tessellations as he mentions the same examples -- the Spanish Alhambra museum and M.C. Escher's paintings -- that appear in the U of Chicago.
My student was able to figure out most of the questions on my worksheet. I noticed that for the example on the front page -- tessellating a triangular region ABC -- my student followed the same steps 1 and 2 as the U of Chicago but then reflected the row over a horizontal axis. This pattern will also fill the plane, but not the same way as the example in the text. It took a while for him to figure out how to tessellate the kite, but he eventually did it. For the opposite -- a tessellation in which no kites occur -- he used a regular hexagon as the fundamental region.
I tried to sing the Square One TV song about tessellations for him -- as much as I could remember. I know that I've already posted this video to the blog, but let me post it again anyway.
But since I'm repeating this video, let me at least do the courtesy of posting the lyrics. These lyrics come from the following link:
http://wordpress.barrycarter.org/index.php/2011/06/07/square-one-tv-more-lyrics/#.VVJkFflVhBc
Afterward I showed him the PARCC Practice Test Question that I am going to post today. Usually, I don't discuss the same question with him that I'm posting the next day, but Question 21 of the PARCC Practice Test is on reflections, so it's relevant to the current Chapter 9 of the Glencoe text:
http://wordpress.barrycarter.org/index.php/2011/06/07/square-one-tv-more-lyrics/#.VVJkFflVhBc
Tessellations
Lead vocals by Larry Cedar
Ooh, tessellations
Ooh, tessellations
Ooh, tessellations
We’re talking about tessellations
Ooh, tessellations
Geometric shape sensations
Ooh, tessellations
Who’s got time for good vibrations?
We’ve got to use our imaginations
Ooh bop-bop ooh
Tessellations
Ooh, tessellations
Geometric shape sensations
Ooh, tessellations
Who’s got time for good vibrations?
We’ve got to use our imaginations
Ooh bop-bop ooh
Tessellations
Let’s decorate my surfboard
With geometric shapes and lines
We can use a few polygons
Gonna make it look so fine
We can start by drawing a grid (bop-bop)
Use it as a guide (bop-bop)
And when we’re done we’re gonna have some fun
We’ll hit the waves and ride! ‘Cause
With geometric shapes and lines
We can use a few polygons
Gonna make it look so fine
We can start by drawing a grid (bop-bop)
Use it as a guide (bop-bop)
And when we’re done we’re gonna have some fun
We’ll hit the waves and ride! ‘Cause
We’re talking about tessellations
Ooh, tessellations
Geometric shape sensations
Ooh, tessellations
Who’s got time for good vibrations?
We’ve got to use our imaginations
Ooh bop-bop ooh
Tessellations
Ooh, tessellations
Geometric shape sensations
Ooh, tessellations
Who’s got time for good vibrations?
We’ve got to use our imaginations
Ooh bop-bop ooh
Tessellations
Let’s use a grid of squares
To tessellate this plane
We’ll lay down different polygons
The shapes don’t have to be the same
We’ll line them up, edge-to-edge
So all the sides will meet
We’ll leave no gaps, no overlaps
Look at how the pattern repeats, yeah
To tessellate this plane
We’ll lay down different polygons
The shapes don’t have to be the same
We’ll line them up, edge-to-edge
So all the sides will meet
We’ll leave no gaps, no overlaps
Look at how the pattern repeats, yeah
We’re talking about tessellations
Ooh, tessellations
Geometric shape sensations
Ooh, tessellations
Who’s got time for good vibrations?
We’ve got to use our imaginations
Ooh bop-bop ooh
Ooh, tessellations
Geometric shape sensations
Ooh, tessellations
Who’s got time for good vibrations?
We’ve got to use our imaginations
Ooh bop-bop ooh
Come on and put a square here
And put an octagon there
And when they fit real tight
We’ll know we’re doing it right
Yeah, yeah
And put an octagon there
And when they fit real tight
We’ll know we’re doing it right
Yeah, yeah
We’re talking about tessellations
Ooh, tessellations
Geometric shape sensations
Ooh, tessellations
Who’s got time for good vibrations?
We’ve got to use our imaginations
Ooh bop-bop ooh
Tessellations
We’re making tessellations
Tessellations
(fade out)
Ooh, tessellations
Geometric shape sensations
Ooh, tessellations
Who’s got time for good vibrations?
We’ve got to use our imaginations
Ooh bop-bop ooh
Tessellations
We’re making tessellations
Tessellations
(fade out)
Afterward I showed him the PARCC Practice Test Question that I am going to post today. Usually, I don't discuss the same question with him that I'm posting the next day, but Question 21 of the PARCC Practice Test is on reflections, so it's relevant to the current Chapter 9 of the Glencoe text:
Triangle ABC is graphed in the xy-coordinate plane with vertices A(1, 1), B(3, 4), and C(-1, 8), as shown in the figure.
Part A
Triangle ABC will be reflected across the line y = 1 to form Triangle A'B'C'. Which quadrant will not contain any vertex of Triangle A'B'C'?
(A) first
(B) second
(C) third
(D) fourth
Part B
What is the y-coordinate of B'?
As I said earlier, I showed this question to my student. He had a little trouble with this question. First he didn't remember what the quadrants were, so I had to show him. Students learn about the four quadrants of the coordinate plane in Algebra I, but this is Geometry -- and quadrants aren't something that many students still remember a year later.
My student was able to locate the line y = 1 correctly -- although he was a little undecided on whether to choose a horizontal or a vertical line before selecting the right mirror. The equations of vertical and horizontal lines often confuse students. I've linked to the Oklahoman teacher Sarah Hagan before -- well, Hagan has posted about the mnemonic HOY-VUX. Here HOY stands for Horizontal line, 0 slope, y = ..., while VUX stands for Vertical line, Undefined slope, x = ...:
But how exactly do we reflect over the line y = 1 anyway? My student drew the mirror and simply counted the number of units above and below the line. This is exactly how the U of Chicago, in the first two sections of Chapter 4, teaches students to reflect over the x-axis. Yet we wonder, is there a purely algebraic way to reflect over the line y = 1, such as switching the sign of the y-coordinate allows students to reflect over the x-axis?
We've already considered 2D isometries and even 3D isometries, but let's make it simpler and consider only one-dimensional isometries. In 1D there are only two types of isometries -- translations and reflections. In 3D a mirror is a plane, in 2D a mirror is a line, and in 1D a mirror is a point. If we call the reflecting point m, then the image of a point x is the point the same distance, but on the opposite side of, m. That is:
x' - m = m - x
Adding m to both sides:
x' = 2m - x
In the same way, we will find out that the image of the point (x, y) over the line y = k has coordinates (x, 2k - y), and we may use this formula when k = 1 to find the images of A, B, and C.
We've already considered 2D isometries and even 3D isometries, but let's make it simpler and consider only one-dimensional isometries. In 1D there are only two types of isometries -- translations and reflections. In 3D a mirror is a plane, in 2D a mirror is a line, and in 1D a mirror is a point. If we call the reflecting point m, then the image of a point x is the point the same distance, but on the opposite side of, m. That is:
x' - m = m - x
Adding m to both sides:
x' = 2m - x
In the same way, we will find out that the image of the point (x, y) over the line y = k has coordinates (x, 2k - y), and we may use this formula when k = 1 to find the images of A, B, and C.
Since A is on the reflecting line, A' is the same as A(1, 1), in the first quadrant. My student figured out that B' has coordinates (3, -2) (in Quadrant IV) and C' has coordinates (-1, -6) (in Quadrant III), so the missing quadrant is the second quadrant, choice (B).
There was one hiccup for my student in Part B. He wrote down 4 instead of -2 -- clearly he thought that the question was asking for the y-coordinate of B rather than B'.
Yesterday, I wrote that hard word problems on tests aren't terrible if they're authentic. Of course, one will be hard-pressed to show that a word problem on reflections is authentic. I've already stated that I prefer reflections over the axes to those over other lines like y = 1, and expecting the students to remember quadrants from geometry doesn't help.
How would I improve this question? I suppose I could change the mirror from y = 1 to the x-axis, but notice now that both A' and B' are in Quadrant IV with no points in Quadrant I. Frankly, I'd probably leave out the quadrants all together. So all that would be left would be to identity the image of one of the points. Perhaps now I'd ask for both the x- and y-coordinates of one of the images, so that students would have to know that one of them changes its sign while the other keeps its sign.
Of course, I might also choose to throw out the reflection question completely and replace it with something else, such as -- tessellations? We are about two-thirds of the way through this practice test and, for all of the traditionalist worries about tessellations, they don't appear on the test, despite their inclusion in Section 9-4 of Glencoe and Section 8-2 of U of Chicago. Then again, tessellations are not very easy to test -- the only tessellation questions in the SPUR section are in the Skills section -- and the skill is to draw a tessellation. This isn't a good question for a computerized test, unless we're going to have the students draw tessellations in LOGO or something.
PARCC Practice EOY Exam Question 21
Yesterday, I wrote that hard word problems on tests aren't terrible if they're authentic. Of course, one will be hard-pressed to show that a word problem on reflections is authentic. I've already stated that I prefer reflections over the axes to those over other lines like y = 1, and expecting the students to remember quadrants from geometry doesn't help.
How would I improve this question? I suppose I could change the mirror from y = 1 to the x-axis, but notice now that both A' and B' are in Quadrant IV with no points in Quadrant I. Frankly, I'd probably leave out the quadrants all together. So all that would be left would be to identity the image of one of the points. Perhaps now I'd ask for both the x- and y-coordinates of one of the images, so that students would have to know that one of them changes its sign while the other keeps its sign.
Of course, I might also choose to throw out the reflection question completely and replace it with something else, such as -- tessellations? We are about two-thirds of the way through this practice test and, for all of the traditionalist worries about tessellations, they don't appear on the test, despite their inclusion in Section 9-4 of Glencoe and Section 8-2 of U of Chicago. Then again, tessellations are not very easy to test -- the only tessellation questions in the SPUR section are in the Skills section -- and the skill is to draw a tessellation. This isn't a good question for a computerized test, unless we're going to have the students draw tessellations in LOGO or something.
PARCC Practice EOY Exam Question 21
U of Chicago Correspondence: Section 4-2, Reflecting Figures
Key Theorem: Definition of reflection
For a point P not on a line m, the reflection image of P over line m is the point Q if and only if m is the perpendicular bisector ofPQ.
For a point P on m, the reflection image of P over line m is P itself.
For a point P not on a line m, the reflection image of P over line m is the point Q if and only if m is the perpendicular bisector of
For a point P on m, the reflection image of P over line m is P itself.
Common Core Standard:
CCSS.MATH.CONTENT.HSG.CO.A.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Commentary: The first two sections of Chapter 4 include several questions where the students must reflect points over the axes. This includes Questions 11 and 12 in Section 4-1 and Question 16 in Section 4-2. In the SPUR section, graphing questions usually appear in the Representations category, where Questions 37-40 are relevant. There are no questions in the text where the mirror is a line other than the axes.
CCSS.MATH.CONTENT.HSG.CO.A.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Commentary: The first two sections of Chapter 4 include several questions where the students must reflect points over the axes. This includes Questions 11 and 12 in Section 4-1 and Question 16 in Section 4-2. In the SPUR section, graphing questions usually appear in the Representations category, where Questions 37-40 are relevant. There are no questions in the text where the mirror is a line other than the axes.
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