"...The universe stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are...geometric figures...without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth."
-- Galileo
Today is my 37th birthday, and as usual, I celebrate it here on the blog in style! First, let's go back to the computer emulator:
http://www.haplessgenius.com/mocha/
Select "Music 6" from the Mount Disk menu, turn on the sound, and then enter the following lines:
LOAD "BIRTHDAY.BAS"
RUN
Also, today my Amazon delivery arrived, and so I now have the Mathematics Calendar 2018 from Pappas this year. It's a sight for sore eyes after she didn't create a calendar for 2017. In order to qualify for free shipping, I added her newest book to the order, Do the Math! This is a puzzle book, and while I could post puzzles from it in the new year, the purpose of writing about Pappas this year is to fill the void created by the lack of a 2017 calendar. Again, the plan is to post only questions from the new calendar that are related to Geometry, just as I sometimes did before 2017.
Last year, my 36th birthday was my ninth Julian birthday. There is nothing special about my 37th birthday, but according to the following link:
https://mrob.com/pub/math/numbers-6.html
two movies revolve around the 37th birthday of their main characters (Phenomenon and To Gillian on Her 37th Birthday).
Last year, on my 36th birthday, I played the Conjectures/"Who Am I?" game. This fit the occasion because the first question in that game is "What is the teacher's age?" And so the students were able to guess my new age on my birthday. (More recently, I wrote about the Conjectures/"Who Am I?" game in my Halloween post.)
I also wrote that I used questions from the STEM project in each grade in the birthday game. In last year's post, I lamented that I should have asked simpler questions to the seventh and eighth graders -- such as how to measure angles -- before jumping directly to the STEM projects where the Illinois State text asked them to make conjectures about the angles they measured. This is part of my general concern with the projects -- that students didn't know how to do what the instructions asked. I should have created worksheets to accompany the Illinois State text.
Indeed, last year my birthday fell on a Wednesday. If I had followed the weekly plan, then STEM projects were for Thursdays, not Wednesdays -- which would be Learning Centers instead. On my birthday, I could have used the Learning Centers as groups for the Conjectures/"Who Am I?" game, and then asked the groups questions about measuring angles. Then this would have prepared them in earnest to do the project on Thursday. (This, of course, assumes that I don't just do a science project on Thursday instead. But even then, the birthday game could have been set up to prepare the students for the science project the next day.)
Seventh graders had music that day last year, and the music teacher sang me "Happy Birthday." But little did I know it at the time, but that was the last time I would ever see the music teacher. The following week was the last school Wednesday of the month and hence an assembly, and then the music teacher was injured over winter break. By the time he healed, the semester ended, and music was replaced with SBAC Prep.\
A few sixth graders quickly drew me a birthday card, and after school during the Wednesday Common Planning meeting, the director (principal) gave me a special birthday basket filled with sweets. But it is bittersweet to reflect on my 36th birthday as I celebrate my 37th birthday today. I wasn't in a classroom today, not even to sub, and so I couldn't play the birthday game with any students at all today. And I learned much too late how to supplement textbook projects with a worksheet so that students can follow them better.
Let's return to Pappas. We read page 37 today since I covered the first thirty pages the corresponding days in November, then read page 31 on December 1st, up to page 37 today. But it also fits perfectly because 37 is my age.
This is the first page of the section "Geometric Worlds." This is a Geometry blog, and so I celebrate my birthday by reading one of my favorite sections of the book.
Here is an excerpt from this page:
"Mathematics has many types of geometries. Each geometry forms a mathematical system with its own undefined terms, axioms, theorems, and definitions."
The one picture on this page is an example of non-Euclidean geometry -- hyperbolic geometry. The caption begins:
"This is an abstract design of Henri Poincare's (1854-1912) hyperbolic world."
Poincare, Poincare -- hey, where have we heard that name Poincare before? That's right -- the same mathematician who formulated the Poincare Conjecture also came up with a model of hyperbolic geometry, the Poincare disc. Indeed, even though we're done with Szpiro's book, let's add the label Poincare in order to write about his disc. The caption continues:
"Here a circle is the boundary of this world. The sizes of the inhabitants change in relation to their distance from the center. As they approach the center they grow, and as they move away from the center they shrink. Thus they will never reach the boundary, and for all purposes, their world is infinite to them."
In past years, I've mentioned hyperbolic geometry as part of "neutral geometry." Since then, I've dropped references to hyperbolic geometry in order to avoid confusing students. If we want to introduce our students any non-Euclidean geometry at all, it's much better to show them spherical geometry than hyperbolic geometry. The world is more like a sphere than a Poincare disc.
Lesson 7-4 of the U of Chicago text is called "Overlapping Triangles." In this lesson students will write more sophisticated proofs.
This is what I wrote two years ago about today's lesson:
Lesson 7-4 of the U of Chicago text covers more proofs. These proofs are trickier, since they involve overlapping triangles.
Because the triangles overlap, it may appear that, in Question 4, we need to show that Triangle SUA is isosceles. But as it turns out, we actually don't need to show this to complete the proof.
The bonus question is somewhat interesting here. It asks whether there is a valid congruence theorem for quadrilaterals, SSASS. Last year I tried to solve it, but got confused, so I want to take the time to set the record straight.
[2017 update: Hey, it's my birthday, so let's convert this into a new multi-day activity -- especially since all the proofs are on the first page, with review on the second page, so we can just replace that second page with this new activity. I added in the Exploration Question from Lesson 7-3, on SSSS, just to put SSASS into perspective -- and it also reflects how I should have set up the projects last year, with a simpler question on the first day before the main question on the second day.]
As it turns out, SSASS is not a valid congruence theorem for quadrilaterals. A counterexample for SSASS is closely related to a counterexample to SSA for triangles -- we start with two triangles that satisfy SSA yet aren't congruent -- one of these will be acute, the other obtuse. Then we reflect each triangle over the congruent side that is adjacent to the congruent angle. Each triangle becomes a kite -- as the original triangles aren't congruent, the kites can't be congruent either, yet they satisfy SSASS (with the A twice as large as the A of the original triangles).
I tried to prove SSASS by dividing each quadrilaterals into two triangles, then using SAS on the first pair and SSS on the second. The problem with this is that that division doesn't produce two triangles unless the quadrilateral is known to be convex. With our two kites, notice that the acute triangle becomes a convex kite, while the obtuse triangle becomes a nonconvex (or concave) kite -- which is also known as a dart. If both quadrilaterals are already known to be convex, then my proof of SSASS is valid.
One congruence theorem that actually is valid for quadrilaterals is SASAS. We can prove it the same way that we proved SAS for triangles. We put one of the sides -- in this case the congruent side that's between the other two congruent sides -- on the reflecting line. Then we can prove that the two far vertices are on the correct ray, the correct distance from the two vertices on the reflecting line -- this works whether the quadrilateral is convex or concave. We can also prove SASAS by dividing the quadrilateral into triangles. There are separate cases for convex and concave quadrilaterals, but all of them work out.
Other congruence theorems for quadrilaterals are ASASA and AASAS. Another congruence theorem, AAASS, is also valid, but it's similar to AAS in that there's a trivial proof based on the angle-sum that reduces it to ASASA (just as AAS reduces to ASA), only in Euclidean geometry. A neutral proof of AAASS exists, but it's more complicated.
[2017 update: I'll retain this reference to "neutral geometry" since I did mention hyperbolic geometry and the Poincare disc in this post. Oh, and if students finish the activity early, it's possible to ask them to solve the Exploration question for Lesson 7-5: Explore this conjecture. If, in quadrilaterals ABCD and EFGH, angles A, C, E, and G are right angles, AB = EF, and BC = FG, then the quadrilaterals are congruent. It turns out that this conjecture is false -- again, a counterexample is a pair of kites, one a square, the other not a square. Then again, if you're tired of giving false conjectures, you can give them one of the valid ones instead like SASAS or ASASA.]
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