Lesson 4-3: Using an Automatic Drawer (Day 25)

Today's scheduled lesson is another technology-based lesson. Just as I did with Lesson 2-3 three weeks ago, I'm supplementing this with an extra worksheet. It's also about graphing -- except this worksheet involves making reflections on graph paper.

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 PP', which means that lines PP' and l have opposite reciprocal slopes. So just to perform the reflection, we need slopes and thus ultimately, dilations. And so we wouldn't be able to work on the coordinate plane until after the unit on dilations.

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:

http://www.geogebra.org/cms/en/

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:

http://mathhombre.blogspot.com/search/label/Geogebra

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.

Lesson 4-2: Reflecting Figures (Day 24)

This is what I wrote last year about today's lesson. Last week I mentioned that I was giving a quiz because it was the end of the "quaver" -- in this post, I explain what a "quaver" is. I had to make one major change to this post -- Dr. Franklin Mason has changed his Lesson 3.1 since I first posted this.

Lesson 4-2 of the U of Chicago text moves from reflecting individual points to reflecting entire figures. The cornerstone is the Reflection Postulate. Nearly every theorem involving transformations and congruence can be ultimately traced back to the Reflection Postulate.

If an academic year is 180 days, then a quarter is 45 days. Then today marks the midpoint of the first quarter according to my calendar. Half of the academic year is called a semester, and half of a semester is called a quarter, but what is half of a quarter called?

Twenty-one years ago was the year that I first learned high school geometry, but it was another class -- history -- where I wondered about half of an academic quarter. This was because my school divided the school year into quarters, and midway through the quarter, unsatisfactory notices were due. And therefore the history teacher gave a major test halfway through the quarter, and again at the end of the quarter. And so I wanted to give a name to the period that was half of a quarter.

The name I came up with is quaver. This term comes from music -- half of a quarter note is known as an eighth note, but the British call eighth notes "quavers." And so today marks the end of the first quaver and the start of the second quaver of my academic year. (Of course I'm referring to my Middle Start district -- today I subbed in a science class in my Early Start district, where today is Day 34.)

Of course, this is all illogical nonsense. In music, a quaver isn't half of a quarter -- it's half of a quarter note, and only when we mix American and British terms. In the UK, a quarter note is called a "crotchet," and so if I really want to be consistent, I should call an academic quarter a "crotchet." But I won't. On this blog, I'm going to call half of a quarter, a quaver. The word quaver sounds a lot like quarter, and I like referring to each quaver of the academic year the same way I refer to each quarter, so there!

Of course, we as geometry teachers aren't always linguistically consistent either. Let's look at the names of various terms for polygons:

n-sided polygon (poly-, Greek "many")
5-sided pentagon (penta-, Greek "five")
6-sided hexagon (hexa-, Greek "six")
7-sided heptagon (hepta-, Greek "seven")
8-sided octagon (octa-, Greek "eight")

That's all fine, except when we reach:

9-sided nonagon (nona-, Latin "nine each")

And so we mix up Greek prefixes with Latin prefixes, just as I mix up the American quarter (note) with the British quaver. The logical term for a 9-sided polygon using a Greek prefix is "enneagon." (Notice that back in Lesson 2-7, in the Bonus question, one of the obsolete terms for a 9-sided polygon was "enneagon.")

Incidentally, I do use musical terms for periods longer than a quaver -- though not "crotchet." In Britain, a "breve" is twice as long as an American whole note, while a "longa" is twice as long as a breve. And so I occasionally refer to periods of two- and four-years as breves and longae, respectively. Thus the nominal length of each of the "paths" that I mentioned a few weeks back is a breve, while the nominal lengths of high school and (undergraduate) college are each a longa. (The official names of these two periods are "biennium" and "quadrennium" respectively -- but I like my names better.)

Now returning to the subject of this blog, 21 years ago -- or is that five longae and one year -- ago, I first sat down in a geometry class. But due to the Common Core, things have changed. Many statements that were postulates 21 years ago are now theorems -- most notably SSS, SAS, and ASA. They are provable statements, and the proofs begin with the Reflection Postulate.

Let me state the Reflection Postulate, because it is so important, directly from the U of Chicago text:

Reflection Postulate:
Under a reflection:
a. There is a 1-1 correspondence between points and their images.
This means that each preimage has exactly one image, and each image comes from exactly one preimage.

b. If three points are collinear, then their images are collinear.
Reflections preserve collinearity. The image of a line is a line.

c. If B is between A and C, then the image of B is between the images of A and C.
Reflections preserve betweenness. The image of a line segment is a line segment.

d. The distance between two preimages equals the distance between their images.
Reflections preserve distance.

e. The image of an angle is an angle of the same measure.
Reflections preserve angle measure.

This postulate corresponds to Dr. Franklin Mason's "Rigid Motion Postulate," in the old version of his Lesson 3.1 last year. Since then, Dr. M has completely changed his Chapter 3 -- this is almost certainly because isometries ("rigid motions") aren't emphasized on the Common Core texts nearly as much as either of us thought they would when we first read the standards. Nowadays, Dr. M uses the classical definition of congruent polygons (i.e., equality of corresponding measures). He assumes SAS as a postulate (just as the mathematician Hilbert did a century ago), and uses Euclid's ancient proof to derive ASA. But for SSS, Dr. M still uses rigid motions to move one of the triangles into place (similar to the start of the U of Chicago proof) before using SAS and Isosceles Triangle Theorem to prove SSS.

Part a is a very important part of the Reflection Postulate. Without it, a point A could have two reflection images -- there could be two distinct points B and C such that the reflecting line m is the perpendicular bisector of both AB and AC. (I believe that the Ruler and Protractor Postulates, or their U of Chicago equivalents, are sufficient to prove the existence of at least one reflection image, but to prove that at most one reflection image exists requires the Reflection Postulate.) This was a problem for Dr. Hung-Hsi Wu, who decided to define rotation before reflection and then use the properties of rotations to prove that every point has at most one reflection image. But since I want to use reflections to define rotation, I am forced to assume that reflection images uniquely exist as part of the postulate to avoid circularity.

According to the text, reflections preserve:
Angle measure
Betweenness
Collinearity
Distance

a nice little mnemonic for the students.

The first theorem of this chapter is the Figure Reflection Theorem:
If a figure is determined by certain points, then its reflection image is the corresponding figure determined by the reflection images of those points.

This theorem is used to conclude, for example, that if A' is the image of A, and B' is the image of B, then A'B' is the image of AB. An informal proof is given. A formal proof is a little bit tricky -- sure, we know that if C is between A and B, then C' is between A' and B'. But the problem is the converse -- if there's a point D between A' and B', how do we know there's a point C between A and B such that C' is exactly D? The best way is probably to use the Distance part of the Reflection Postulate -- we choose C to be the point on AB such that AC = A'D (which exists by Ruler Postulate). Then since reflections preserve distance, C' must be exactly D.

But that is the sort of proof that I don't want to confuse students with. It's best just to use the informal proof given in the text and save formal proofs for later.

The text states that when a figure intersects the reflecting line, the image must intersect the reflecting line in the same point or points. This follows immediately from the fact that the image of a point on the reflecting line is the point itself.

But what I find interesting is the related statement -- if a figure intersects its reflection image, then it must intersect the reflecting line in the same point or points. This statement is false in general, but it's true if the figure to be reflected is itself a line. This fact helps us greatly -- for example, consider Question 21 from the text:

The reflection image of Triangle ABC is Triangle XYZ. Now Lines AC and XZ intersect at a point -- which we now know must lie on the reflecting line. And Lines BC and YZ intersect at a point -- which we now know must lie on the reflecting line as well. And those two points determine exactly one line -- the reflecting line! So all the student has to do is draw the line through the two points of intersection.

And as a corollary, it follows that if a line is parallel to the reflecting line, it must be parallel to its reflection image, Last year, I called this the "Line Parallel to Mirror Theorem" (where "mirror" refers to the reflecting line) But I will wait a few days before introducing that theorem to students. (Notice that the Common Core Standards state that a line must be parallel to its dilation image, so why not give the conditions when a line is parallel to its reflection, rotation, or translation images?)

Lesson 4-1: Reflecting Points (Day 23)

This is what I wrote last year about today's lesson. It is all about reflections:

At last, we have reached what makes Common Core Geometry different from traditional geometry -- transformations, including reflections, rotations, and translations. So far, what I posted in September is not much different from a traditional course. But I had to give all that preliminary material first -- after all, the Common Core Standards demand it:

CCSS.MATH.CONTENT.HSG.CO.A.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

And that's exactly what had to do in the first three chapters of the U of Chicago text --define angles, circles, perpendicular lines, parallel lines, and line segments. Only now after defining those basic terms can we actually define rotations, reflections, and translations so that we can finally do Common Core Geometry.

Lesson 4-1 of the U of Chicago text deals with reflections. As I mentioned last year, we do reflections first because the text defines rotations and translations in terms of reflections!

The definition of reflection is so important that I repeat it here. (Remember that I use a strikethrough to represent the segment symbol, since I can't reproduce the vinculum here.)

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 PQ.
For a point P on m, the reflection image of P over line m is P itself.

This definition is highly intuitive -- after all, suppose I gave a student a line m and two points P and Q such that the reflection image of P over m is the Q. Now suppose that I drew in segment PQ, and asked the student to tell, just by looking, how many degrees is the angle formed between m and PQ. Chances are that the student will say that it is 90 degrees. Then suppose that I asked the student to tell me which of P and Q is closer to m. Chances are that the student will say that the distances are the same.

(This is a trick that I often do with students -- whenever I ask a student whether AB or CD is longer, I'm almost always trying to get the student to notice that the lengths are equal!)

For this section, I'll repeat my first worksheet on reflections. Then I follow it with some exercises. Keep in mind that the method I suggested to generate reflection images is folding -- and it may be hard to fold when there is writing on both sides. As much as I want to save paper and not tie up the Xerox machine, this lesson, and the ones that follow, are very intensive on drawing and folding.

Still, there are a few more things that I want to include here. As I mentioned earlier, one way to generate reflection images is folding. Another method suggested in the U of Chicago text is utilizing a protractor. And once again, this is an important lesson, so let me restate the method for those of you who don't have the U of Chicago text:

1. Place your protractor so that its 90-degree mark and the center of the protractor are onm.
2. Slide the protractor along m so that the edge line (the line through the 0- and 180-degree marks) goes through P.
3. Measure the distance d from P to m along the edge line. You may wish to draw the line lightly.
4. Locate P' on the other side of m along the edge, the same distance from P.

We can see why this works. The first two steps take care of the "perpendicular" part of the definition -- as m is on the 90-degree mark and PP' is on 0 and 180 degrees, so m is perpendicular to PP'. Then the last two steps give us the "bisector" part -- with P and P' both d units from m, so that m bisects PP'.

This text gives a third method -- constructing using a straightedge and compass. But I wish to save constructions for a little later.

Notice that I prefer just to use the prime symbol for the image of P, giving us P'. The U of Chicago text also gives function notation, r(P). I admit that this is more precise -- especially if we, just as the book does, add a subscript to indicate the line of reflection. But once again, my priority is to avoid confusing students, and I hope that students will find P' less confusing than r(P).

The text also includes reflections on the coordinate plane over one of the axes. I decided to include this, since we'll have to use a coordinate plane at some point in this course. But I still want to avoid writing linear equations until we get to the second semester (at which point the Common Core Standards have us do something brand new with equations). Since reflecting individual points on the plane has nothing to do with equations, I have no problem including this here.

I had to throw out most of the review questions for this lesson, since most of them pertain to the second half of Chapter 3 that I've skipped for now. Only Question 22 can be included -- and even then, only the (a) part fits here. Even though the question is labeled (Lessons 3-3, 3-2), the (b) part requires parallel lines and corresponding (actually same-side interior) angles, so it's actually part of Lesson 3-4.

The final Bonus Question 24 is actually a preview of the next lesson, Lesson 4-2.