Monday, June 23, 2014

One Resource a (Week)Day #2: Interesting Mathematics from 1970!

Today's resource of the day is also an old publication, and one that I have seen on more than one school's shelf. I'm pretty sure I saw this book back at my school in El Salvador, but never quite looked through it then. I picked it up immediately this summer when I saw it in the to-be-donated stack at school because of this odd familiarity. The publication date is 1970, and it reads like it, because the textbook, entitled Mathematics - A Human Endeavor and authored by Harold R. Jacobs, has the tagline on the inside cover, "A Book for Those Who Think They Don't Like the Subject" and is laid out in such a way that it overviews topics from across the spectrum of algebra, geometry, statistics, etc. It's not like a modern textbook. It doesn't bother with test preparation. The author is interested in authentic mathematics that generates genuine interest, and it really shows.

The book reads more like the mathematical storybooks that my parents had bought me as a child. The table of contents looks like this (I picked out the chapters that I particularly liked, but it's most of the book):

Ch. 1. Mathematical Ways of Thinking, introduces inductive thinking through billiard logic and deductive thinking through proving number tricks.
Ch. 2. Number Sequences, covers a variety of sequences (arithmetic, geometric, cubic, etc.) all the way through the Fibonacci Sequence, each with interesting applications.
Ch. 3. Functions and Their Graphs, starts to look at the connection between equations, predictions, and graphs but doesn't go into nitty gritty details of each form.
Ch. 4. Large Numbers and Logarithms, introduces logs and exponents in historical perspective, as well as real applications (beyond earthquakes and decibels).
Ch. 6. Mathematical Curves, introduces classic geometric shapes via their construction methods. Including spirals and cycloid! (which often gets omitted in modern textbooks since they're not immediately useful to the standardized "bottom line.")
Ch. 7. Ch. 8. Ch. 9 build up an essential understanding of basic counting, probability, and statistics, tying in interesting examples as how to apply statistics to decipher secret codes!
Ch. 10 Topics in Topology, starts with a brief exploration of networks, shortest paths, and ends with questions about the Moebius Strip. 

Anyhow. This book is amazing. It's a math-lover's book as much as it is accessible for those who perhaps don't already love math. To give you just a small taste of the brilliance of this book, here are some nice bits when I was flipping through the book.

This is a sequence problem that visually represents Fibonacci Sequence. It made me think about how much trickier it would be to construct a geometric representation of f(n) = f(n - 1) + f(n - 2) + f(n - 3).

This is the classic historical view of logs. I saw this at PCMI once, and it's pretty much presented the same way here. They give log tables in the book as well, for use in the exercises.

I think this is the first time I have seen a math problem about submarine cables.  It's an awesome application of scientific notation, especially because  I think our students probably don't realize that there are fiber-optic backbones to the internet and it can potentially generate all kinds of interesting questions.

I have seen billiard problems, but I liked these. These were different from the ones I have seen. Assuming that the initial shot goes out at 45 degrees, you can ask many interesting questions if you know the length and width of the table and can assume the ball to be of point size. 

I hope you've enjoyed today's edition of One Resource a (Week)Day! Stay tuned for more tomorrow!

PS. Incidentally, my old school in the Bronx used Elementary Algebra also by Harold R. Jacobs. That's a really nice algebra 1 and 2 textbook. Not fancy but solidly usable, with great writing and some nice problems/explanations.

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