"Determine how to cut this shape into three pieces with two straight cuts, so that the three pieces can be rearranged into a square."
This is the final page of the section "Some Mathematical Recreations," and our last mathematical recreation is called "The Square Transformation." Once again, we have a completely visual puzzle, and so I have no choice but to describe it here in words.
The figure fits in a rectangle of base 18 and height 27. But the upper-left corner -- a square of length 9 -- has been removed. So we are technically left with a hexagon, alternating between vertical and horizontal sides of length 27, 18, 18, 9, 9, 9.
As usual, I'll post the complete answer tomorrow. But let's look for some hints. We notice that the figure consists of five squares of length 9, and so its area is 405 square units. A square with the same area must have sides of length sqrt(405) or 9sqrt(5) units. So somehow our cuts must lead to sides of this irrational length. This strongly implies that the cuts are neither horizontal nor vertical, but are diagonal (or oblique). A cut across the diagonal of a 2 * 1 rectangle is sqrt(5) units, and so our cuts will ultimately have to be nine times this length. That's all I'll say about this until tomorrow!
Meanwhile, here are the solutions to last week's doublets puzzle:
(1) EYE-LYE-LIE-LID
(2) PITY-PITS-FITS-FINS-FIND-FOND-FOOD-GOOD
(3) TREE-FREE-FLEE-FLED-FEED-WEED-WELD-WOLD-WOOD
(4) OAT-RAT-ROT-ROE-RYE
You can see how this works. EYE to LID was straightforward -- just change the first letter, then the second letter, then the third letter. But OAT to RYE doesn't work as easily -- RAT is a word, but RYT is not. But at least RAT gives us a start -- and remember the hint last week about vowels, so at some point we needed a word with two vowels (ROE). The four-letter words were indirect -- it's interesting that in both cases we had intermediate words beginning with F.
We didn't cover much in the Pappas recreation section -- mainly because half of the section was blocked by the weekend and three straight non-blogging days. But I do wish to write about one of the skipped pages, 238.
In the past on the blog, I've mentioned that October is Hexaflexagon Month. The month of October was selected because it's the birthday of their inventor, Martin Gardner. Notice that Gardner has appeared in many of my recent posts since both Pappas and Hoffman keep mentioning the famous recreational mathematician in their books. But Gardner's actual birthday is the 21st -- a Saturday this year, hence it's yet another non-posting day blocked by the weekend.
Anyway, on page 238, Pappas writes about something called a "hexatetraflexagon." In this case, the prefix "hexa-" implies that the "toy," if you will, has six faces, but each face has only "tetra-," or four sides (a square), rather than the usual six.
Sarah Carter, one of the most famous math teacher bloggers, would write an annual post about hexaflexagons in her classroom. Even though Gardner's birthday is in October, it would usually be November or December by the time she writes the post. But she didn't write about hexaflexagons in 2016 at all. Instead, here's a link to her November 2015 post:
https://mathequalslove.blogspot.com/2015/11/hexaflexagon-day-2015.html
So I'll certainly be on the lookout for her 2017 hexaflexagon post the next two months! Meanwhile, the queen of the hexaflexagon is Vi Hart. Here's a link to one of her videos:
You may be asking, is there a video about hexatetraflexagons, since that is what actually appears on Pappas page 238? The answer is yes -- here's a video by Jill Britton:
That's good -- I won't have to type in the instructions from page 238, since you can just watch the video above.
OK, so let's get back to Paul Hoffman's book. On Friday, we read two chapters in order to cover the extra Chapter e. Today is Chapter 3 -- and naturally, if there's a Chapter e, there's also a chapter numbered for the other famous transcendental number, pi. So once again we have two chapters. Just like Friday, I'll probably just summarize the chapters and skip over large portions of them in posting about them on the blog.
Chapter 3 of Paul Hoffman's The Man Who Loved Only Numbers is "Einstein vs. Dostoyevsky." As usual, the chapter begins with a quote:
"My own greatest debt to Erdos arises from a conversation 30 years ago in the Hotel Parco del Principi in Rome...."
Wow, I usually like to post the complete opening quote, but this one's quite long (two paragraphs). I already told you that I'm skipping today, so let's skip to the end of the quote:
"Ever since then, I've realized that I'm infinitely rich: not just in the material sense that I have everything I need, but infinitely rich in spirit in having mathematics and having known Erdos."
-- Richard Guy
By the way, Richard Guy is a British mathematician. As of today's post, he is still alive -- last month he celebrated his 101st birthday! So if you want the full story of his meeting with Erdos, ask him!
Chapter 3 opens in 1959 when Erdos is allowed to enter the U.S. for a math conference -- in other words, he catches (Uncle) Sam in a good mood. On his return to Hungary, he meets a twelve-year old prodigy, Louis Posa, whose mother wants him to meet the famous mathematician.
And so Erdos asks him, "Prove that if you have n + 1 integers less than or equal to 2n there are always two of them which are relatively prime." Hoffman gives us the following example:
"As an example, choose n to be 5. Then the conjecture is that if you take any six integers from the set 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, you can't avoid choosing two that are relatively prime."
Hoffman tells us Posa's response -- "The two are neighbors." As it turns out, this is yet another example of the Pigeonhole Principle that I mentioned in my last post. We divide the 2n natural numbers into n pairs -- {1, 2}, {3, 4}, {5, 6}, all the way up to {2n - 1, 2n}. Since there are n pairs and n + 1 numbers to choose, two of them must be in the same pair -- and so the GCD of the two of them must be 1.
Ultimately, Erdos homeschools the young "epsilon" Posa. When the youngster is 14, he gains the Erdos number of 1 as he and the famous mathematician write a paper together. Hoffman tells us that Posa enjoys algebra, but doesn't care for calculus or -- gasp -- geometry. (The gasp is because this is a geometry blog!) The chapter title comes from Posa, who announces that he'd rather be Dostoyevsky, the Russian novelist, than Einstein. This is at the age of 20, when he becomes an elementary teacher.
The young Posa is also distracted by girls, and so he asks Erdos why there are so few girl mathematicians around. Naturally, Erdos responds with a story about "slaves" and "bosses." He says, "Suppose the slave children would be brought up with the idea that if they are very clever, the bosses will not like them. Would there then be many boys who do mathematics?"
Here Erdos implies that the reason for the lack of female mathematicians is that guys say that they are turned off by smart women. This, of course, is the image that Danica McKellar tries to fight. I also mention this in my February 21st post, when some of the eighth grade girls in my class aren't eager to learn math. My support provider dispels the notion that guys are turned off by intelligent women, and then I bring out McKellar's book.
Later on in this chapter, Hoffman writes about another time when Erdos and his mother are allowed to enter the United States. They stay with Vazsonyi, a childhood friend and fellow mathematician, right here in Southern California -- Manhattan Beach, to be precise. Hoffman reminds us again that attended school too -- who tries to make her friendship with Erdos more than platonic anyway. Of course, the mathematician rejects her advances.
The only woman Erdos is ever close to is his mother. This is why he is very depressed when she dies of an ulcer in 1971. Hoffman closes the chapter by telling us how from that day on, Erdos spends 19 hours a day doing math, in order to cope with his mom's passing.
Chapter pi of Paul Hoffman's The Man Who Loved Only Numbers is "Dr. Worst Case" As usual, the chapter begins with a quote:
"Dear Ron, When Paul Erdos's mother died, someone told me that she was the last of his living relatives...."
Really, Hoffman -- the opening quote is an entire letter! Let's skip again to the end:
"And to you, who have invested more than any person in smoothing his way in this rough society, I express my sympathy and my thanks for all you have done. With my sincerest good wishes, Gordon Raisbeck."
The author of this letter is, of course, another mathematician acquaintance. But our focus is the person to whom the letter is addressed -- Ron Graham. Hoffman mentions Graham earlier in this book, but now Hoffman begins Chapter pi by writing about Graham's story.
Graham is born right here in California -- albeit Taft, in Northern California. (He is still alive -- his 82nd birthday is coming up on Halloween.) Hoffman tells us that Graham first becomes interested in math when he memorizes numbers on his paper route, and his fifth grade teacher helps him use this mental ability to calculate square and cube roots.
His seventh grade teacher poses a problem -- find the size of a population of rats if the rats die at a rate proportional to the size of the population -- and the older man uses this problem to motivate Graham to learn calculus independently. He ultimately graduates high school at 15 and is awarded a scholarship to attend the University of Chicago. (Hey, that sounds familiar, since of course I write about the U of Chicago text all the time.)
Three years later, Graham's father convinces him to transfer from "the dangerous, leftist" U of Chicago to "an all-American school" like UC Berkeley. (This is ironic from a 2017 perspective, considering the political images of those schools now. But recall that this was the 1950's.)
Hoffman tells us that Graham's Ph.D. dissertation is on unit fractions. These fractions go all the way to the ancient Egyptians, who wrote all fractions as the sum of distinct unit fractions. Graham's dissertation is all about how to convert fractions most efficiently. For example, he writes:
3/7 = 1/3 + 1/11 + 1/231
3/7 = 1/6 + 1/7 + 1/14 + 1/21
3/7 = 1/4 + 1/7 + 1/28
Which of these is the "best" -- the first one (greedy algorithm, where we have the largest possible fraction 1/3), the second one (smallest maximum denominator 21), or the third (fewest number of terms, 3, with a small maximum denominator)?
I could keep going on with more of Graham's discoveries in this chapter -- but I really do need to get to the textbook that comes from Graham's original college, the U of Chicago.
Lesson 4-3 of the U of Chicago text is called "Using an Automatic Drawer." The newer Third Edition of the U of Chicago text diverges wildly from my old Second Edition -- after Lesson 4-2, the lessons don't really line up again until Chapter 7. Many lessons from Chapters 4, 5, and 6 appear are placed in a different order between the Second and Third Editions. Indeed, Lesson 4-3 of the Third Edition is the same as Lesson 6-4 of the Second (on miniature golf and billiards), and in general the rest of the new Chapter 4 is the old Chapter 6 (on the other transformations -- translations and rotations). On the other hand, the old Lesson 4-3 doesn't appear in the new text at all (due to updates in technology).
This is what I wrote two years ago 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 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.
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 2015 (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. [2017 update -- his most recent post now has a similar survey. Euclid is still up there, joined now by Pythagoras.]
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