Anyway, this school ordinarily has a block schedule, but this week it's mixed up. That's because yesterday there was no school (hence no blog post), and tomorrow is PSAT day. Ordinarily, both Monday and Wednesday are odd days, with even Tuesday and Thursday and all-classes Friday. But I knew coming in that we wouldn't want to have two even days with no odd days, and so I arrive at school knowing that today isn't going to be a normal even day.
To my surprise, today is the all-classes day. Since Thursday is already even day, that leaves Friday to become the odd day. (I was actually expecting today to be odd, so that Thursday could remain even and Friday all-classes as usual. I suspect the reason for making today all-classes is that if it were odd instead, then even classes would go six days -- from last Friday to this Thursday -- without meeting.)
And apparently, the regular teacher doesn't know that today is an all-classes day either. The lesson plan that he leaves me lists only the even blocks (and writes that it's OK for me to release the students for "embedded support" -- but there is no embedded support on all-classes days).
So in a way, I was once again without a lesson plan for the odd classes. Fortunately, this teacher was also out last Friday, so I have some idea as to what these classes are doing (similar to the situation where the teacher is out Thursday-Friday but only leaves a Thursday lesson plan). Fifth period, an advanced drama class, has something to rehearse, while third period beginning drama can be given the same lesson plan as the even periods 2 and 6. That leaves first period -- which isn't even a drama class, but senior English. On Friday they did something on Chromebooks -- but then those laptops were delivered to another classroom (since they aren't usually kept in the drama room). The teacher never requested them for today because he thought that odd classes don't meet today.
So instead, I give this class a free period. I try to avoid free periods in middle schools (with my "Who Am I?" game), but with more mature seniors, a free period usually isn't a problem.
Originally I don't plan on playing songs today, but the teacher has a guitar in his office, and so I can't resist playing it. In both first and fifth periods, I play four songs -- "U! N! I! T! Rate," "Earth, Moon, and Sun," "One Billion Is Big," and "Meet Me in Pomona."
Unfortunately, his guitar has only five strings -- the high E is missing. You'd think that this might cause a problem in playing E chords, but it doesn't -- an open E chord goes EBEG#BE, so omitting the high E leaves EBEG#B, still a credible E chord. The problem is actually in playing D chords -- an open D chord is typically played as DADF# on the top four strings. Omitting the high string leaves only DAD -- what's called a power chord with no major third. In certain situations, adding back the low A makes a pleasant ADAD power chord, but not when the song is actually in the key of D.
(It's also possible to use drop-D tuning, but I don't dare tune someone else's guitar -- and besides, we'd be left with a DADAD power chord. But now it might be possible to fret the D-string at F# to make a DAF#AD chord for D major. Once again, though, I don't tune another person's guitar.)
The drama classes watch a video -- an episode of "Doctor Who." Once again, the teacher has assumed that today would be a block day, with plenty of time to watch the video and answer the questions about the video in their journals. Instead, with shorter periods, there's barely enough time to finish the video. It goes without saying that I don't sing in these classes, nor is there any real criteria to place anyone on the good list.
(In case you're curious, the episode is "Blink." Like many episodes of "Doctor Who," this episode involves time travel. I know it's from the 2007 version of the series because 2007 is the "present year" in the time travel scenes. I wish that I could send a message back in time to the regular teacher, informing him what today's bell schedule is!)
But there are a few problems with defiance, which I tell the regular teacher directly -- he returns to the classroom as soon as school is out. In second period, one student is watching an unrelated TV show on a cell phone, and in sixth period, a group of students is talking in a corner (where they're not supposed to be) and paying zero attention to the "Doctor Who" video. Even though the bell has rung, two students haven't left yet -- and they help me identify the troublemakers to the teacher.
At the time of the incident, I ask the students to move away from the corner. One of them claims that I'm interrupting the video, so I should just return to the teacher's area. This happened a few times at the old charter school -- a student is not paying attention, then claims that I'm the one who's interrupting the lesson. This usually indicates that my rule-explaining has crossed the line into argument, so the best thing to do is end the argument and simply write down what's happening (or give the punishment, if I'm the regular teacher).
During fourth period conference, I'm actually sent to a math class -- Algebra II. But as it turns out, this class is taking some sort of district assessment -- and it is taken on the SBAC interface. This is something new to me -- using the online SBAC interface for district assessments. (Yes, I can't help but think about the SBAC Prep and "dry runs" back at the old charter school.)
The teacher admits that he wouldn't have scheduled the test for today if he'd remembered that he has a meeting for a special ed student (which he was informed about a month ago) -- after all, district tests typically have windows, several days when the test can be given. He's late to the meeting since he must spend several minutes giving the students Chromebooks and all the passwords they need to access the online test. Fortunately, the meeting is quick, and the teacher returns before class ends.
The strand for the test appears to be either SSE (Seeing Structure in Expressions) or APE (Arithmetic with Polynomials & Rational Expressions), with more emphasis on the earlier standards that are covered in high school Algebra I. The teacher tells the students that the district will complain if the Algebra II students can't pass Algebra I standards.
(Also, why in the world would the district schedule the testing window for this week when there's another big test, the PSAT, coming up?)
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
If the diameter of this sphere is sqrt(15/pi), its surface area?
It's clearly a question for Lesson 10-9 of the U of Chicago text, "The Surface Area of a Sphere." We notice that the radius is (1/2)sqrt(15/pi) and the formula is SA = 4pi r^2. We can just look at it and see that the ^2 cancels the sqrt, the 4's cancel (after the 2 is squared to 4), and the pi's cancel, leaving us with 15. The desired surface area is 15 square units -- and of course, today's date is the fifteenth.
(Notice that we could have saved a step by using SA = pi d^2 instead -- then we see the ^2 and sqrt cancel, and the pi's cancel immediately.)
Meanwhile, last weekend was the biannual library book sale. Lately, this is held on the second Saturday in October. So as usual, let me discuss the math books that I purchased this year.
As usual, I purchase one math text. This year, I choose the teacher's edition of an Algebra I text by McDougal Littell. This text is special to me -- it's the Algebra I text that I myself once used as a young seventh grader (except, of course, I didn't have the teacher's edition).
Let's start with the table of contents:
- Introduction to Algebra
- Working with Real Numbers
- Solving Equations and Problems
- Polynomials
- Factoring Polynomials
- Fractions
- Applying Fractions
- Introduction to Functions
- Systems of Linear Equations
- Inequalities
- Rational and Irrational Numbers
- Quadratic Functions
It's been so long since I've seen my old text that I've forgotten that the order of topics is dramatically different from that of the more common Glencoe text. I typically think of factoring as a second semester topic, but in this text it appears in the first semester. On the other hand, graphing doesn't appear until the second semester.
I'm not sure what the most difficult topic in Algebra I is, frankly. I've always remembered my Algebra I teacher -- the one who taught out of the text I now hold in my hands -- as she referred to factoring as "the F-word of Algebra I." But now as a teacher, I've seen so many students struggle with graphing that I wonder whether it's better to follow the order of this text -- teach factoring in first semester, graphing in second semester. (Indeed, I recall that Sarah Carter -- back when she still taught Algebra I -- following this order more closely than Glencoe's.)
The principal author is Richard Brown, but the second author is Mary Dolciani. It's easy to spot Dolciani's influence on the text -- ever since the golden age, she always included "oral exercises" and "written exercises" in each section. This edition is dated 2000 -- five years after Dolciani's death. But the oldest version is dated 1990, just in time for me to have used this text. (I took Algebra I about a year or so before Dolciani passed.) Since the name Dolciani carries weight with traditionalists, this text might be one of the latest-written texts that they would find acceptable. (Of course, we know that they also appreciate "proper Glencoe textbooks.")
If we wished to follow this text yet adhere to the Glencoe order, we might start with Chapters 1-3 and then skip to Chapters 8-10 to round out the first semester. (Today starts the second quarter, so we'd probably jump to Chapter 8 today.) Then we'd return to Chapter 4. Glencoe would probably place the quadratics chapter earlier, before Chapter 6 on (algebraic) fractions. Indeed, the chapters on fractions and radicals (part of Chapter 11) occur late in the Glencoe text -- and in fact, many teachers don't even teach these chapters fully. The Brown/Dolciani text is different in that it saves quadratics for the very last chapter.
In fact, I still remember to this day that the Quadratic Formula (Section 12-3) was the very last lesson that we studied. The teacher had a poster on her podium -- it showed the cartoon character Garfield thinking about the Quadratic Formula. And then she hurried through Chapter 12 in order to get to this lesson during the final lesson, just a day or two before the test on the last day of school. (That school, which spanned Grades 7-12, didn't adopt a finals week until recently.)
Notice that Section 12-3 isn't the last lesson of Chapter 12 -- Section 12-8 ("Joint and Combined Variation") ends the text. I see that on one of the secret teacher pages (one I never saw as a young student) is a pacing guide. The "minimum course" indeed ends with Section 12-3, while the "average course" goes up to Section 12-8. I'm not sure whether my teacher meant to give us the minimum course, but she probably just fell off the pace and it turned out that way. (Then again, the class was for eighth graders with one seventh grader -- yours truly -- so maybe she did want to slow down to the minimum course.)
On the other hand, the maximum course has the Chapter 12 test with three weeks to go, and then adds several "Looking Ahead" sections (including some Geometry and Trig). Notice that some of these "Looking Ahead" sections include probability and statistics -- some of which might be necessary parts of the Common Core Algebra I course here in California.
One page that I still remember to this day is page 524. It appears after Section 11-4, "Irrational Square Roots" -- and is a proof that sqrt(2) is irrational. Until then, I've always heard it asserted that sqrt(2) is irrational (indeed, it's implied in Section 1-4 proper), but I never knew that this was something that could be proved, until the day I saw page 524. This is called an "Extra" (just like "Exploration" in the U of Chicago text). I reckon that most students in my class just ignored this page, but I was fascinated by it.
Here is the proof of the irrationality of sqrt(2), as given by Brown/Dolciani:
- Assume that 2 has a rational square root.
- Then, sqrt(2) = a/b, where a and b are positive integers that have no common prime factorl that is, a/b is in simplest form.
- If sqrt(2) = a/b, then 2 = a^2/b^2. Since a^2 has the same prime factors as a, and b^2 has the same prime factors as b, a^2 and b^2 have no common prime factors. Thus, a^2/b^2 is in simplest form.
- Multiplying both sides of the equation a^2/b^2 = 2 by b^2, you have a^2 = 2b^2. Thus, a must be even because its square is even. (Recall that the square of an even integer is even and that the square of an odd integer is odd.)
- Since a is even, you can write a = 2n for some integer n. Then, substituting 2n for a in a^2 = 2b^2, you have (2n)^2 = 2b^2, 4n^2 = 2b^2, or 2n^2 = b^2.
- Since 2n^2 = b^2, b must be even because its square is even. Therefore, you may write b = 2m for some integer m.
- Therefore, both a and b have 2 as a factor. This contradicts the fact that a and b have no common prime factor.
- Hence the assumption that 2 has a rational root is false since it leads to a contradiction. QED
Exercise
Prove that sqrt(3) is irrational.
At the book sale, there was also a student copy of the Algebra II text by Brown/Dolciani. But I only purchased the teacher copy of Algebra I. I believe that I might have studied out of the Algebra II text as a freshman, but then in November I moved to a different district. That district used the U of Chicago text for Algebra II instead (which I already bought at a previous book sale).
There are other books I purchased at the book sale on Saturday. One of them is Ian Stewart's The Story of Mathematics: From Babylonian Numerals to Chaos Theory, dated 2007. I've bought books by Stewart before, and this is the sort of text that I might do our fall side-along reading with.
But there are two more surprises waiting for my at the book sale. One of them is a DVD that is part of the Great Courses, Changes in Motion: Calculus Made Clear. The teacher is Professor Michael Starbird of the U of Texas at Austin. It's the second Great Courses DVD that I own -- the first is David Kung's course on paradoxes. When I first received Kung's course, I did a side-along reading (or viewing) of it on the blog.
Finally, I bought a text book on coding -- Sams Teach Yourself Visual Basic 6 in 24 Hours. The Cal State University system is on the verge of requiring a fourth year of math for college entry -- but computer science can count as one of the years of math. I feel that this is the way the winds are blowing now -- I'm obviously having trouble getting hired as a math teacher. Requiring more math to get into Cal State might increase the demand for math teachers -- but if one of those years can be comp sci, then it increases the demand for coding teachers as well.
I'm not quite sure whether Visual Basic is the language of choice, though. I know that Java is now the language for the AP Comp Sci exam, and other languages such as Python are growing as well. I do know that Visual Basic is the modern version of BASIC (the language mentioned in the U of Chicago text and used by the Mocha emulator). Surely Visual Basic is more useful than the BASIC spaghetti code that I've already learned. There is a CD-ROM which allows me to download and actual program in Visual Basic, so it's more useful than a book on another computer language that I can't actually program in. I'm not a coder until I write actual code that makes an actual computer work. I need practice, not theory. But one way to force me to learn the lessons is to make it as part of side-along reading, so that I must read and blog about the experience. The book is dated 1998, so I do wonder whether the CD will work on a modern computer.
So now I have three solid choices for our next side-along reading -- the Stewart book, the Starbird Calculus course, and VB coding. I'll make my decision in time for tomorrow's post, since that will be the first side-along reading post for the fall.
This is what I wrote last year about today's lesson:
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 ofPP', 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.
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.
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
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.
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.
[2019 update: Of course by "BASIC" here I mean spaghetti-code BASIC, not Visual Basic.]
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.
2019 Update: Math Hombre still posts from time to time, although he now teaches Algebra II rather than Geometry. His most recent post is dated June 28th:
http://mathhombre.blogspot.com/2019/06/playful-math-129.html
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.
2019 Update: Math Hombre still posts from time to time, although he now teaches Algebra II rather than Geometry. His most recent post is dated June 28th:
http://mathhombre.blogspot.com/2019/06/playful-math-129.html
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