doing math with a cas lin mcmullin usacas conference 2007
TRANSCRIPT
DOING Math with a CAS
Lin McMullin
USACAS Conference 2007
Cubic Symmetry
Show that any cubic polynomial has a point of rotational symmetry.
q p x q p
q p q p x
,p q p
p x p x
Cubic Symmetry
Analytic Geometry
Analytic Geometry
Perpendicular bisector theorem:
Investigate the set of all points (x, y) in a plane equidistant from P(6,1) and Q(–5,7). Find the
a) Equation of points equidistant from P and Q.
b) Equation of the line between P and Q
c) Intersection of the lines
d) Midpoint of segment PQ
Analytic Geometry
A Locus problem:
Graph the set of points, (x, y) the difference of whose distance from and is 3 2,3 4, 1
Trigonometry
SSS triangle.
64.5
8
4.5cos ,4.5sin
8,0
2 24.5cos 8 4.5sin 0 6
Trigonometry
SSA triangle.
This approach can be used for SAS as well.
37.8
68.75
,0c
8.75cos 37.8 ,8.75sin 37.8
2 2
8.75cos 37.8 8.75sin 37.8 0 6c
Trigonometry
ASA triangle.
15
50.743.5
ba
cos 43.5a cos 50.7b
cos 43.5 cos 50.7 15
sin 43.5 sin 50.7
a b
a b
The Trapezoid Problem
• A trapezoid with base 1 = a, and base 2 = b. Draw a segment that is parallel to the bases and divides the trapezoid's area A into A1 and A2. Represent the length of the segment in terms of a and b if A1 = A2.
a
b
c
x
h x
h1A
2A
The Trapezoid Problem
11 2
1 2
A A
A A
11 2
1 2
A A
A A
1 1 12 2 2
1 12 2
a c x a b h
a c x c b h x
2 2 2 2
2 2
2
2 2
2 2
2
a b a bc
a a b c ax h h
a b b a
a
b
c
x
h x
h1A
2A
Altitudes in a Right Triangle
Given a right triangle with legs of a and b, express the lengths of the segments , in terms of a and b
Geometry Expressions
Altitudes
1 2 3, , , , nh h h h
a BC
AB
C
D
E
b
FH
GI
1h2h
3h4h
Altitudes
2
2
22 2 2 2
2 2 222 2
2
2 2 2 2
ab aba
a ba b a bh
a bab ab
aa b a b
How is DOING Math Different with a CAS?
• The CAS does the algemetic so we can concentrate on the mathematics.
• You can improve the CAS by adding your own operations and routines.
• New approaches are possible once you stop worrying about the algemetic.
• “Go for the equation.”
• Complicating can make the work go faster.
• One still needs to know mathematics.
Implications for teaching
Good CAS use is a new skill, a new tool that students must be taught and encouraged to learn.
To do this we needA willingness to accept new ways of doing problems
A new style of showing work
A change in how we think about “simplifying”
A good source of better problems for students to attempt
DOING Math with a CAS
The text of this presentation along with the slides, examples and solutions are available at
www.LinMcMullin.net
Click on “Resources” then on “CAS”
DOING Math with a CAS
Lin McMullin
USACAS Conference 2007