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Showing posts from September, 2008

3.2 Descartes and the Coordinate Graph

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This lesson leads off with an intro to Rene Descartes the 17th century philosopher who is credited with coming up with the idea of coordinate geometry (not surprisingly called the Cartesian coordinate system.) There's a cool, but most likely apocryphal story of how Descartes discovered the idea. My quick Google search came with this description:
One morning [], Descartes found himself watching a fly on the wall (or so the story goes) and suddenly discovered that he could define the fly’s position using only three numbers: the perpendicular distance of the fly from each wall and from the ceiling. Generalizing from this realization, he discovered that any point in space could be defined in a similar way by measuring their distances from perpendicular lines or planes. These numbers have commonly become known as “Cartesian coordinates” and the perpendicular lines as the x- and y-axes. That discovery led to the development of analytical geometry, the first mathematical blending of a…

What are functions?

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Wikipedia.com defines a function in mathematics as an abstract entity that associates an input to a corresponding output according to some rule.

A good mental model for this idea is a function machine. My colleague at CIESE (where I worked until last October) Jason Sayres created this nice example which he calls The Mystery Box Game* which is really version of "Guess my Rule". For example, let's say

2 goes through the box and becomes 5
5 goes through the box and becomes 11
9 goes through the box and becomes 19
Can you predict what rule is being used here? (Spoiler alert: click here)

A rule is a mathematical way of explaining a pattern if one exists. Since mathematics can be thought of as the study of patterns, rules or functions are foundational for understanding mathematics.
Hopefully, the rest of lesson 1 (Chap 3) will be easier to understand. (Note: Bob has posted the answers for the first lesson of the first edition on our Jacobs group site.)

So what's …

Re: Running late - phew!! - an update

Karen writes:
Don't know about anyone else but I, too, am not having much luck at getting to the chapter [3] so far. I am delighted that you are running late!!

Ihor replies:
May I suggest you put a bookmark wherever you are and join us in Chapter 3 if that's OK. One of the things I really like about HE (Human Endeavor) is that each chapter potentially could be done separately though I'm pretty sure Harold didn't intend that since he was writing it for high school teachers in conventional classrooms. I think a lot about Web 2.0 and math and I'm sure that the direction we're heading towards is short chapters on topics that can stand alone. Or I should say short STORIES rather than chapters since chapters usually implies a long, boring textbook.

I'm rethinking my timeline for Chapter 3 since this is my favorite chapter and I hope you will join us in participating in the culminating activity which is the Great Green Globs Contest. (Sounds perfect for Hallo…

Chapter 3 - Functions and their Graphs - Kickoff

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After an all too short, but hopefully productive hiatus known as summer, I thought I would bring us back to Harold Jacob’s world and his fascinating look at mathematics, which indeed is a human endeavor.

Last year we completed the first two chapters of his book where we explored mathematical ways of thinking and number sequences (chapters 1 and 2.) In Chapter 3 we’ll take a look at the powerful idea of functions, their graphs and the stories they tell.

Since I’ll act as your guide for at least this next chapter I’ll try to capture what made Mr. Jacob’s book (including his teacher’s guides) so special in my growth as a teacher. More about that as we go along.

Here’s the lesson outline:

Functions and their Graphs
1. The idea of a function
2. Descartes and the coordinate graph
3. Graphing linear functions
4. Functions with parabolic graphs
5. More functions with curved graphs
6. Interpolation and Extrapolation: Guessing Between and Beyond
Chapter 3/Review/Problems for further…