Problems for CAS Solution

Lin McMullin

MATH & SCIENCE TECHNOLOGY CONFERENCE

January 18, 2008Norman, Oklahoma

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

A Cubic’s Roots

Show that the tangent line to a cubic at the point where x = the average of two of its roots, intersects the cubic at its third root.

Ratios, We got Ratios

Ratios, We got Ratios

1511

7

Analytic Geometry

Analytic Geometry

Perpendicular bisector theorem: Investigate the set of all points (x, y) in a plane equidistant from P(–3,2) and Q(5,4). Find the

a) Find the length of

b) Write the equation of the perpendicular bisector of

c) Write the equation of the set of points such that the sum of the distances from A and B is 9.

d) Graph the locus found above.

AB

AB

Analytic GeometryGiven the quadrilateral with vertices

( 5,2), (11.3,7.1), (16.4,5.0) and (0.1, 0.1)A B C D

a. Show that ABCD is a parallelogram.

b. Are the diagonals perpendicular? Show how you know.

c. Show that the diagonals bisect each other.

TrigonometrySSS 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 28.75cos 37.8 8.75sin 37.8 0 6c

Trigonometry

ASA triangle.

1550.743.5

ba

cos 43.5a cos 50.7b

cos 43.5 cos 50.7 15

sin 43.5 sin 50.7

a b

a b

Quartic Points of Inflection

,b q b

,a q a

?,?

?,?

Where else does the line through the points of inflection of a fourth degree polynomial intersect the polynomial?

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

Problems for CAS Solution

Lin McMullin

MATH & SCIENCE TECHNOLOGY CONFERENCE

January 18, 2008Norman, Oklahoma

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”

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

cx

h x

h1A

2A

The Trapezoid Problem 1

1 2

1 2

A AA A

11 2

1 2

A AA 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 ha b b a

a

b

cx

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

1h2h3h

4h

Altitudes

2

2

22 2 2 2

2 2 222 2

2

2 2 2 2

ab abaa ba b a bh

a bab aba

a b a b