Math 5270 Transformational Geometryemina/teaching/5270s13/5270_Day2_notes.pdf · Transformational...

Math 5270Transformational Geometry

Day 2

Summer 13

Day 2, Summer 13 Math 5270 Transformational Geometry 1/43

What is the purpose of geometry in high school education?

Visualization

Spatial reasoning

To get ready for trig

To get ready for calculus

To understand more geometrically the algebra taught

It’s beautiful


Unusual view

Algebra is the delivery system for geometry; I really like geometry

and love how algebra can be used to explore geometry more deeply.


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Where we left off

Predict what these might be doing:

f (x , y) = (x3, y3)

g(x , y) = (2x , 3y)

h(x , y) = ( 3√x , ey )

i(x , y) = (cos x , sin y)

j(x , y) = (−x , x + 3)

l(x , y) = (x3 − x , y)

m(x , y) = (3y , x + 2)

n(x , y) = (x + 2, y − 3)

p(x , y) = (0.6x − 0.8y , 0.8x + 0.6y)

q(x , y) = (−0.6x + 0.8y , 0.8x + 0.6y)

For each of the given transformations, describe their effect on

points of the plane, coordinate axes, and characteristics they leave

invariant.



Observations



Students’ misconceptions

Congruence transformations are the only ones.

Transformations only move a given figure.

To map a segment it’s enough to find images of its endpoints.



Lines

Thales allows us to define slope



Equation of a line

Suppose we’re given a line with a slope a which passes through the

point C = (0, c). For any point P = (x , y) on the line we have:



What came first: chicken or an egg?

Descartes viewed equations of lines as information deduced from

Euclid’s axioms. But we could do it in the opposite direction! Use

equations to define lines (gasp!) and then you can deduce some of

Euclid’s axioms, eg:

there is a unique line through any two distinct points,

for any line ` and point P outside `, there is a unique line

through P not meeting `.



Distance

Pythagora tells us what it ought to be:

Or: we define it then Pythagora comes for free....



Angles: tricky business... Hello, trigonometry.

How would you find the angle between a line y = tx and the

x-axis?

Use relative slope

±∣∣∣∣ t1 − t21 + t1t2

∣∣∣∣to test equality of angles.



Drama Club: Going Round in Circles

Alice: Mike

Lewis Carroll:

Diane

Dr. Whatif:

Chris

Narrator:

Kaitlyn


Math 5270 Transformational Geometryemina/teaching/5270s13/5270_Day2_notes.pdf · Transformational...

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