"Behind the wall, the gods play; they play with numbers, of which the universe is made." -- Le Corbusier (1887-1965)
The is the first page of a new section, "The Arch -- Curvy Mathematics." The opening quote is by a famous Swiss architect who clearly understood the connection between his craft and mathematics.
Pappas begins:
"The arch is an elegant architectural triumph."
She then follows this with a long list of arches. Six of these are pictured on the page:
-- semicircular arch
-- horseshoe arch
-- stilted arch
-- pointed arch
-- ogee arch
-- elliptical arch
The following is a link to some of these arches. The arches appear on the fifth and sixth slides:
https://www.slideshare.net/kavin_raval/arches-61544375
Today we begin Chapter 2 in the U of Chicago text. It's also a great day to begin the side-along reading book that I purchased last month, Excursions in Number Theory, a Dover book written by C. Stanley Ogilvy and John T. Anderson in 1966. I know, it's odd to have a side-along reading book when we're already reading Pappas on the side. Still, I'm stretching out Pappas over several months, while I'll read Ogilvy's book a chapter a day.
Here is the table of contents:
1. The Beginnings
2. Number Patterns
3. Prime Numbers as Building Blocks
4. Congruence Arithmetic
5. Irrationals and Iterations
6. Diophantine Equations
7. Number Curios
8. Prime Numbers as Leftover Scrap
9. Calculating Prodigies and Prodigious Calculations
10. Continued Fractions
11. Fibonacci Numbers
As you can see, our reading of Ogilvy's book will line up almost exactly with Chapter 2 in the U of Chicago text. And so without further ado, let's begin.
Chapter 1 of Stanley Ogilvy's Excursions in Number Theory is called "The Beginnings." He writes:
"In the beginning there were no numbers; or if there were, primitive man was unaware of them. Whether the numbers were always 'there' (where?), or had to be invented, has been a much discussed question, and we shall leave it to philosophers to continue that discussion without our aid."
Interestingly enough, Le Corbusier (from today's Pappas quote) actually gives an answer. Let's repeat his quote from above:
"Behind the wall, the gods play; they play with numbers, of which the universe is made." -- Le Corbusier (1887-1965)
So the answer is that the numbers were always there, since "the gods" had numbers well before man was aware of them. And Le Corbusier's answer to "where?" is the entire universe.
Ogilvy asks us to wonder what the addition table for schoolchildren would look like before there were numbers -- if the only quantities we had were "none," "few," and "many." He writes:
none + none = none
none + few = few few + few = ?
none + many = many few + many = many many + many = many
Many students who dislike math may wish that this is our addition table! But the author tells us that counting was developed just to answer that one unknown question -- when is few plus few no longer few, but many?
At this point, Ogilvy compares the multiplication of whole numbers to that of polynomials. To multiply 307 * 43, he writes:
(3x^2 + 7)(4x + 3) = 12x^3 + 9x^2 + 28x + 21.
We now substitute in x = 10, and we obtain 13,201 -- the answer to the whole number problem. In fact, we often point this out in Algebra I classes when we teach polynomial multiplication. As he proceeds to explain:
307 = 3 * 10^2 + 0 * 10^1 + 7 * 10^0, where 10^0 means 1.
The reason that this works is that we have a decimal, or base 10, numeration system. Ogilvy uses this idea to introduce other number bases, particularly the binary system used in computers. (I wrote about this more in my July 22nd post, during the Pappas chapter on computers.)
Ogilvy shows us how to convert a number from decimal to binary. His example is 22, and he writes:
22 0
11 1
5 1
2 0
1 1
The first column is the number to be converted, and each number in the column is half of the number above it (either exactly or discarding the half). The numbers in the second column are determined from those in the first column -- zeros are to the right of even numbers and ones are to the right of odd numbers. The binary digits appear in the second column, starting from the bottom, so 10110 in binary is the same as 22 in decimal. He tells us that:
22 = 1 * 16 + 0 * 8 + 1 * 4 + 1 * 2 + 0 * 1 or 10110.
Ogilvy tells us that this binary notation is very closely related to peasant multiplication. This is a nonstandard algorithm that depends on doubling and halving. He writes:
Multiplication Division
14 11
28 5
112 1
154
Notice that the crossed-out rows correspond to zeros (discards) in the binary conversion earlier, while the rows to be added correspond to ones. So this is just a clever way to convert one of the numbers to binary before adding. The entire operation can be completed using stones, just as Napier used markers to multiply in binary on a chessboard (as explained in another July post). I doubt that even Common Core advocates would propose this method as an alternative to the standard algorithm.
Ogilvy's final example is in Roman numerals. In this example, he adds 49 to 162. He writes 49 as XXXXVIIII (as XLIX would be awkward here) and 162 as CLXII.
XXXXVIIII
CL X II
CC X I
According to the author, "After that, the only thing to do is list letters in suitable columns and be careful about carrying."
One thing about number theory, the subject of this book, is that it's the form of higher math that is accessible to the most people. "Homology theory" depends on knowing what a homology is, "field theory" depends on knowing what a field is, "category theory" depends on knowing what a category is, and so on. But "number theory" requires us only to know what a number is. This is why I can choose a number theory book for side-along reading. Of course, no matter which theory we choose, we must understand definitions -- and that is the subject of today's Geometry lesson.
Lesson 2-1 of the U of Chicago text is called "The Need for Definitions." This is what I wrote two years ago about today's lesson. The part about the definitions of words from outside of mathematics -- such as terrorist -- is even more timely today as the blog calendar has placed this lesson closer to the anniversary of 9/11.
The second chapter of nearly any high school geometry text discusses the logical structure of geometry -- to prepare students for proofs. This includes the U of Chicago text, as well as Dr. Franklin Mason's text, and many others.
Lesson 2-1 of the U of Chicago text deals with definitions. But the introduction to the chapter mentions a 1986 USA Today article concerning a non-mathematical definition: cookie. Normally, as teachers we'd ignore this page and skip directly to the first lesson, except that this article is mentioned all throughout 2-1, even including the questions!
Now, of course, a teacher could have the students discuss the article as an introduction to the importance of precise definitions. Such an introduction is often called an anticipatory set -- a concept that apparently goes back to the education theorist Madeline Hunter.
A teacher could present the article as an anticipatory set, but I should point out that the article is over a quarter of a century old -- after all, my text itself is nearly that old. The article points out that the word terrorist was controversial even back then. As we already know, a decade after the book was written, the 9/11 attacks occurred -- and since then, that word terrorist has been thrown around so much more, with very strong political implications.
And, of course, there was another definition that led to a politically charged debate -- one that occurred just a few years after the publishing of the text. During the investigation during the impeachment of Bill Clinton, the former president questioned the definition of the word is. So we see that there are two fields where precise definitions matter greatly: law and mathematics.
To me, it might be fun to discuss these examples in class. But it may be tough for the teacher to remain politically neutral during such a discussion, so we must proceed with caution.
The images at the end of this post do not mention the article -- I threw out any part of the section that refers to the article. I preserve the discussion about what a rectangle is, and the one definition given in the lesson -- that of convex set.
When approaching the questions, I first threw out Questions 1 through 3, since these questions go back to the article. I kept all of the questions about convex sets, since that's the term defined in the lesson, then kept the question where students guess the definition of midpoint -- a preview of Lesson 2-5.
Now I want to consider including the review questions as well. As any teacher knows, students have trouble retaining what they've learned, so we give review questions to make them remember. I avoided review questions during Chapter 1 since most of them were review of the skipped Lessons 1-1 through 1-5. But most of the review questions in this section are labeled Previous course. I must be careful about these, since it all depends on which previous course is being mentioned here.
Question 20 in the text discusses the definition of words like pentagon and octagon. Like midpoint, this book will define these terms later in the chapter (Lesson 2-7), but this is labeled Previous course. I assume that the intended previous course is probably a middle school course. But -- remembering that this is a Common Core blog -- I decided to look up the Common Core Standards. The only standard mentioning the word pentagon is a 2nd grade standard!
CCSS.MATH.CONTENT.2.G.A.1
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
So in theory, it might have been nearly a decade since the students saw the word pentagon. (The word octagon doesn't appear in the standards at all.) But I figure that upon seeing the question, the students will remember vaguely that these words all refer to shapes with different numbers of sides -- and at least know that a triangle has three sides, even if they must guess on all the rest. This is a good preview of Lesson 2-7.
In the other questions marked Previous course, the course referred to is clearly Algebra I. Once again, I don't want to intimidate the students with Algebra I questions in a Geometry class. Of course, we can see how Questions 21 and 22 came about -- they are clearly translations of the word problems "23 degrees less than the measure of an angle is the measure of its supplement" and "the measure of an angle is six times the the measure of its complement," respectively. I'm torn whether to include such problems. One thing that I definitely want to avoid is algebra problems masquerading as geometry problems -- for example, we take a linear equation from algebra and write its two sides as the measures of vertical angles (provided the two sides equal valid angle measures). The geometry in such a question is trivial -- just set the two sides equal to each other since vertical angles are congruent, then the rest is all algebra. The geometry in a question about complementary and supplementary angles is less trivial, but then -- so is the algebra, since a typical question will often have variables on both sides, and many students struggle with these.
In the end, I decided to keep Questions 21 and 22 but at least give the students a break by making the solutions whole numbers -- notice that as written, the solutions to both contain fractions. Question 23 seems to serve no geometric purpose at all. I decided to drop the second variable z and change the number 225 to 360, since students will often need to divide 360 degrees by various numbers -- for example, when finding the exterior angle measures of a 15-gon. This is the most difficult algebra/arithmetic that I want appearing in the first semester of a geometry course -- nothing beyond this is acceptable.
Finally, we reach Question 24. This is an Exploration question, asking the students to define the words cookie and terrorist. Once again, this makes a lot more sense if the article is mentioned in class. I decided that I'll include this and other Exploration questions, but label them as Bonus questions to emphasize that these questions are optional for the students. Of course, it can be thrown out completely if a teacher wants to avoid politically charged debates over the word terrorist.
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