Math 5270
Transformational Geometry
Day 2
Summer 13
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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
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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
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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
Predict what these might be doing:
f (x, y ) = (x 3 , y 3 )
g (x, y ) = (2x, 3y )
√
h(x, y ) = ( 3 x, e y )
i(x, y ) = (cos x, sin y )
j(x, y ) = (−x, x + 3)
l(x, y ) = (x 3 − 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.
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Observations
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Students’
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.
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Lines
Thales allows us to define slope
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Equation
ofwith
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:
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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 `.
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Distance
Pythagora tells us what it ought to be:
Or: we define it then Pythagora comes for free....
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Angles:
tricky
Hello,
trigonometry.
How would you find the angle between a line y = tx and the
x-axis?
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Angles:
tricky
Hello,
trigonometry.
How would you find the angle between a line y = tx and the
x-axis?
Use relative slope
t1 − t2 ±
1 + t1 t2 to test equality of angles.
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Drama Club: Going Round in Circles
Alice: Mike
Lewis Carroll:
Diane
Dr. Whatif:
Chris
Narrator:
Kaitlyn
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