The relationship between the coordinate plane transformations -- including reflections -- in Common Core Geometry is a bit complex. On one hand, many of the properties of the coordinate plane, such as the slopes of parallel and perpendicular lines, depend on dilations and similarity -- and we know that this is emphasized in the standards. This ultimately affects reflections on the plane -- suppose we have the coordinates of a point P and the equation of a line l, and we wish to find the coordinates of P', the reflection image of P. Now by the definition of reflection, line l is the perpendicular bisector of
But on the other hand, reflections are easier for students to visualize -- and therefore understand -- if students can draw them on the coordinate plane. This is especially true for the simplest mirrors, namely the x- and y-axes. We don't need to know anything about slope in order to perform reflections over the coordinate axes. And indeed, there's a brief reference to such reflections over the axes on my Lesson 4-1 worksheet.
Yet this isn't nearly enough emphasis on the coordinate plane when we consider the Common Core exams such as PARCC and SBAC. Of the four questions on the PARCC Practice Exam that mention reflections, three of them take place on the coordinate plane. As usual for the blog, the PARCC exam takes priority over all other considerations. My duty on this blog is to make sure that students are prepared to do well on the Common Core exams.
The reflections that appear on the PARCC usually have one of the coordinate axes as a mirror, but we've also seen other horizontal and vertical mirrors, as well as y = x and y = -x as mirrors. It can be argued that one doesn't really need dilations or slope to reflect over horizontal or vertical mirrors, provided we take it for granted that any horizontal line is perpendicular to any vertical line and that we can easily find distance along a horizontal or vertical line.
Is it possible to prove that the reflection image of (x, y) over y = x is (y, x) without having previously to prove anything about dilations or slope? On one hand, it may seem that we could prove that the line y = x forms a 45-degree angle with either axis simply by showing, for example, that (0, 0), (x, 0), and (x, x) are the vertices of an isosceles right triangle. Then the line y = -x also forms a 45-degree angle with the axes, and so the angle between y = x and y = -x must be 45 + 45, or 90, degrees. And so we can show that the lines y = x and y = -x are perpendicular, which is a start.
And all of this, of course, requires us to prove that the graph of y = x is even a line! (Interestingly enough, today I subbed in an art class where the students were learning the concept of line. Art defines the word line differently from geometry -- according to a video featuring several famous artists, a point is a dot, and a line is a dot that moves. A line in art can be any shape, even a circle. I noticed that one artist in the video was using software that looked very similar to the Geogebra program that I mention later in this post.)
But even after proving that the equation of y = x really is linear, we'd still need to find distance along the the oblique lines y = x and y = -x, and this seems to be impossible without having a Distance Formula, which comes from the Pythagorean Theorem, which in turn comes from similarity and dilations. So it indeed appears impossible to show that the reflection image of (x, y) is (y, x) before the similarity chapter.
And so I've decided to create a worksheet just with reflections over the coordinate axes. I've added on a "reflection square" from last year, which students can fold to see the reflections.
This is what I wrote last year about today's lesson on the graphing calculator. Notice that instead, one can try my new worksheet, where students reflect points over the axes, on a graphing calculator too:
Lesson 4-3 of the U of Chicago text is another technology chapter. But this time, the text describes something called an "automatic drawing tool" or "automatic drawer."
Now I like including technology sections, since these show to the students that geometry isn't just something done in the classroom, but is actually performed out in the real world. But the last time there was a technology chapter -- Lesson 2-3 -- I converted the BASIC programs given in the U of Chicago text into TI-BASIC programs for the graphing calculator. But this section will be more difficult, precisely because the TI-83 or TI-84 is not an automatic drawer. The TI was designed to graph functions and equations -- in other words, do algebra. It was not designed to measure distances, and especially not angles -- in other words, do geometry. So many of the tasks described in the text are not doable on the TI.
As it turns out, there does exist an online graphics program that performs both geometry and algebra -- appropriately enough, it's called Geogebra:
I'm not familiar with Geogebra, since I've never downloaded it on used it in a classroom. But based on what I've heard about it, Geogebra can perform all of the tasks described in Lesson 4-3. Much of what I know about Geogebra I read on the blog of John Golden, a mathematics professor from Michigan who calls himself the "Math Hombre." Here's a link directly to the "Geogebra" tag on Golden's blog:
One thing I learned about Geogebra is not only can it reflect figures over a line -- which is of course the topic for the current chapter -- but it can reflect figures over a circle as well! A circle reflection is not, however, one of the transformations required on Common Core. But I think that it's interesting to compare circle reflections to the Common Core transformations, just in case someone sees that option on Geogebra and wants to know what a circle reflection is.
As you might expect, a circle reflection maps points inside the reflecting circle to points outside the circle, and vice versa -- and just as with line reflections, the image of a point on the reflecting circle is the point itself. Preimage points close to the center of the reflecting circle have points that are far away from the center -- indeed, halving the distance from the preimage to the center ends up doubling the distance from the image to the center. This means that if the preimage is the center itself, its image must be infinitely far away. It's a special imaginary point called "the point at infinity."
A circle reflection is definitely not an isometry -- that is, the Reflection Postulate certainly doesn't hold for circle reflection. Part b of that postulate states that the image of a line is a line. But circle reflections don't preserve collinearity. As it turns out, though, the image of a "line-or-circle" is a "line-or-circle" -- if the preimage line passes through the center, then its image is itself, otherwise, the image ends up being a circle.
My favorite part is what happens when we find the composition of two circle reflections. As we will find out later in the U of Chicago text (and as I mentioned last year), the composition of two reflections in parallel lines is a translation. Well, the composition of two reflections in two concentric circles happens to be -- a dilation! And just as we can easily find the direction and distance of the translation -- its direction is perpendicular to the two reflecting lines, its distance is double that between the two lines -- we can find the center and scale factor of the dilation. The center of the dilation is the common center of the two reflecting circles, while the scale factor is the square of the ratio of the radius of the second reflecting circle to that of the first. (So the dilation is an enlargement if the second circle is larger than the first and a reduction if the second circle is smaller than the first.)
But let's return to the TI. For the sake of those teachers who have access to TI in the classroom, but not Geogebra, let me make Lesson 4-3 into a lesson fit for the TI-83 or TI-84. Here are some commands that will be helpful for drawing on the TI. (Before beginning the following, make sure that there are no functions turned on under Y=.)
First, we'll usually want to turn the axes off for this. So we press 2nd FORMAT (which is the ZOOM key) to choose AxesOff. If we press GRAPH, the screen should be blank. If it isn't, we press 2nd DRAW (which is the PRGM key) to choose ClrDraw. Many of the following commands can be found on this 2nd DRAW menu.
The command Line( draws a line -- segment that is. The arrow keys and ENTER are used to select the starting and ending points. We can also draw an individual point by moving to the right of the DRAW menu to the POINTS menu and choosing Pt-On(.
Now we're in the reflection chapter, so I want to bring this back to reflections. Unfortunately, the TI doesn't automatically reflect for us. So the students will have to reflect instead. One way of doing is to divide the class into partners, and give a calculator to each pair. Then one partner can draw the preimage triangle, and the other add the image onto the picture. Example 2 on the U of Chicago text may be awkward, though, since the reflecting line is oblique (that is, neither horizontal nor vertical), s one might want to try a horizontal or vertical reflecting line first before trying an oblique line.
Example 4 is especially nice. The first partner can draw triangle ABC first, then the second partner can reflect it to draw triangle ABD, and then the first partner takes the calculator back to draw both triangles CEF and DEF.
Interestingly enough, a question in the text that's very suitable for TI drawing is Question 22, in the Exploration (or Bonus) section of the Questions. Part a -- a spiral made up of straight line segments -- is extremely easy to draw on the TI. One can use the Line( command to draw each segment, or even use the Pen command (choice A, the final choice on the Draw menu). After selecting Pen, all the student has to do is press ENTER at the beginning of the spiral, then move with the arrow keys until reaching the end of the spiral, then pressing ENTER again.
Part b is more of a challenge, though. Since this picture contains circles, the Circle( command (choice 9 on the Draw menu) will come in handy. Notice that the endpoints of all the segments in the picture are either points on the circles or centers of the circle. Because the picture has reflectional symmetry, this is also a good picture for drawing from the command line. The necessary commands happen to be Line(X1, X2, Y1, Y2) to draw a line segment from (X1, X2) to (Y1, Y2), and Circle(X,Y,R) to draw a circle with center (X,Y) and radius R. If a student uses this method, it will be a good idea to make the viewing window symmetrical and square by choosing ZSquare or ZDecimal from the ZOOM menu. (I personally prefer ZDecimal, since it makes the pixels correspond to integers and multiples of .1, which is easier and also makes the graphs more accurate.)
On my worksheet, I give some simple commands for TI drawing, then move on to the Exercises based on the Questions in the book. For simplicity, I decided to keep Questions 1-7, but they are reworded to so that they work in classrooms with Geogebra, TI, or no technology at all (where today's lesson would be simply a second day of Section 4-2).
First, Questions 1-2 ask about automatic drawers. Since technically TI is not an automatic drawer, I changed these to simply ask about graphing technology. In a classroom without technology, the students can be made aware of graphing technology without actually using it.
Questions 3-4 involve measuring with a ruler and drawing by hand. So these can be completed in any of the classrooms I described earlier.
Questions 5-7 ask to use an automatic drawer like Geogebra. Classes with TI or no technology can just do these problems by hand like Questions 3-4.
Then I include three review problems that can be completed in any classroom. Finally, I included Question 22 as a Bonus, since these can be completed on either Geogebra or TI. Since it's a bonus question, classes without technology can just ignore this one.
By the way in case you're wondering, Math Hombre's blog is still active. Although his most recent Geogebra post is dated last November (where he shows how to draw various grids on this software), he's written other material since. His last post is all about a college course he's teaching called "The Nature of Modern Mathematics." Notice that he surveyed his students, and when he asked them to name five milestones in the history of mathematics, the two most common answers (with seven responses each) are Geometry and Euclid.