Enjoying geometry proofs
Here recently I finished reading the two Dr. Math geometry books. (I will be writing a review of them, and I can say they're pretty good & inexpensive books!)
The one meant for high school geometry had as its last chapter circles and theorems about circles. Well circles never were the strongest part of my mathematical knowledge, for whatever reason... (I think it stems from the fact that so big part of school geometry concentrates on calculating areas and volumes, and not on properties of figures.)
So I wanted to brush up on circle theorems. I quickly read over the circle chapter on another geometry book I have, called "Geometry: A Guided Inquiry". (I can recommend that book as well. It often asks the student to explore and try find theorems. See more info here.)
I don't know about you, but to me, reading & learning theorems and proofs can be enjoyable. First you maybe struggle to grasp it all, but afterwards there is a great satisfying feeling and admiration of the logical process - and it's BEAUTIFUL.
So I thought I'd blog about inscribed angles... so you see a tiny glimpse of this logical process of proof.
1) Let's suppose we've already proved this theorem: that inscribed angles are equal, and are half of the corresponding central angle.
All these angles A are equal (or congruent as they say), and are half of angle B.
I can even draw many of them in the same pic: now the red, blue, black, and green angles have the same measure, exactly half of angle B.
(As a sideline, if we make B to be 180 degrees so it forms a diameter, then what is the measure of A?)
For proof of this theorem, see Inscribed and Central Angles in a Circle: What is this about? A Mathematical Droodle - a nice interactive applet shows the angle relationship, too.
2) THEN, after understanding that, let's consider two chords in a circle that intersect.
Filling in some more lines, we get what I'd call the "slanted timeglass figure".
We see there are two inscribed angles here, from the same points, and they're congruent (angles A and A').
Angles B and B' are congruent since they're vertical angles.
SO.... even the third angles of these two triangles must be equal (since angles in a triangle always add up to 180 degrees).
It follows that we have two SIMILAR TRIANGLES! We have just PROVEN that the time glass figure inside a circle consists of two similar (same shape) triangles.
Well that was just a tiny tiny glimpse, but I hope you see how this process could work. And I wish our students can get these tiny glimpses even BEFORE high school geometry - to learn appreciate math a little more, to see its beauty in action, to learn what proving is all about. In my mind, proofs are NOT something boring - they're part of the beauty of mathematics.
Tags: math, geometry
The one meant for high school geometry had as its last chapter circles and theorems about circles. Well circles never were the strongest part of my mathematical knowledge, for whatever reason... (I think it stems from the fact that so big part of school geometry concentrates on calculating areas and volumes, and not on properties of figures.)
So I wanted to brush up on circle theorems. I quickly read over the circle chapter on another geometry book I have, called "Geometry: A Guided Inquiry". (I can recommend that book as well. It often asks the student to explore and try find theorems. See more info here.)
I don't know about you, but to me, reading & learning theorems and proofs can be enjoyable. First you maybe struggle to grasp it all, but afterwards there is a great satisfying feeling and admiration of the logical process - and it's BEAUTIFUL.
So I thought I'd blog about inscribed angles... so you see a tiny glimpse of this logical process of proof.
1) Let's suppose we've already proved this theorem: that inscribed angles are equal, and are half of the corresponding central angle.
All these angles A are equal (or congruent as they say), and are half of angle B.
I can even draw many of them in the same pic: now the red, blue, black, and green angles have the same measure, exactly half of angle B.
(As a sideline, if we make B to be 180 degrees so it forms a diameter, then what is the measure of A?)
For proof of this theorem, see Inscribed and Central Angles in a Circle: What is this about? A Mathematical Droodle - a nice interactive applet shows the angle relationship, too.
2) THEN, after understanding that, let's consider two chords in a circle that intersect.
Filling in some more lines, we get what I'd call the "slanted timeglass figure".
We see there are two inscribed angles here, from the same points, and they're congruent (angles A and A').
Angles B and B' are congruent since they're vertical angles.
SO.... even the third angles of these two triangles must be equal (since angles in a triangle always add up to 180 degrees).
It follows that we have two SIMILAR TRIANGLES! We have just PROVEN that the time glass figure inside a circle consists of two similar (same shape) triangles.
Well that was just a tiny tiny glimpse, but I hope you see how this process could work. And I wish our students can get these tiny glimpses even BEFORE high school geometry - to learn appreciate math a little more, to see its beauty in action, to learn what proving is all about. In my mind, proofs are NOT something boring - they're part of the beauty of mathematics.
Tags: math, geometry
Comments
This 'no matter what' part may sometimes be lost (not understood) by the students.
Well-planned lessons with interactive geometry software (Cabri, Geometer's Sketch pad come to mind) can also help with this.