Integrating Computer Algebra Systems into Algebra and Precalculus Courses

1
Integrating Computer Algebra Systems into Algebra
and Precalculus Courses

• Michael Buescher
• Hathaway Brown School

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A Test Question - Algebra 2

• Given an arithmetic sequence a with first term t
  and common difference d,
• Show that a6 a9 a3 a12
• Show that if m n j k,
• then am an aj ak

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What are Computer Algebra Systems?

• Computer-based (Mathematica, Derive, Maple) or
  Calculator-based (TI-89, TI-92, HP-48, HP-49)

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What are Computer Algebra Systems?

• Computer-based (Mathematica, Derive, Maple) or
  Calculator-based (TI-89, TI-92, HP-48, HP-49)
• Allow Symbolic Manipulation

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What are Computer Algebra Systems?

• Computer-based (Mathematica, Derive, Maple) or
  Calculator-based (TI-89, TI-92, HP-48, HP-49)
• Allow Symbolic Manipulation
• Capable of solving equations numerically and
  algebraically

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My Experience

• Using CAS in Algebra 2 and Precalculus classes
  for five years
• TI-89 for all, Mathematica for me
• Traditional curriculum, heavily influenced by
  College Board AP Calculus

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Decision to use the TI-89

• Some students already had it
• More students wanted it
• College Board allowed it for SAT and AP
• Telling adolescents they cant do something is
  always an effective strategy see session 128 on
  Dress Code

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What is your current attitude about Computer
Algebra Systems?
It gives lots of people new life in mathematics.
It lets them focus more on the problem-solving
aspects rather than the tedious computations."
-- James Schultz, Ohio University
This is madness. They won't learn algebra. It
will cut off careers in many fields." -- Richard
Askey, University of Wisconsin at Madison
Madness
New Life
Quotes From Lisa Black, Robert Channick. New
Algebra Batteries Required Chicago Tribune,
October 29, 2003 http//www.chicagotribune.com/new
s/local/chi-0310290205oct29,1,3428295.story
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Construction vs. Education

• You can build a road using shovels and
  wheelbarrows.

• You can build a road using a bulldozer.

Kutzler, Bernhard. CAS as Pedagogical Tools for
Teaching and Learning Mathematics. Computer
Algebra Systems in Secondary School Mathematics
Education, NCTM, 2003.
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Construction vs. Education

• Technology allows us to do some things more
  quickly or more efficiently.
• Technology allows us to do some things we
  couldnt do at all without it.

BUT!
People need to be trained in how to use it!
Kutzler, Bernhard. CAS as Pedagogical Tools for
Teaching and Learning Mathematics. Computer
Algebra Systems in Secondary School Mathematics
Education, NCTM, 2003.
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Transportation vs. Computation
Appropriate Technology
Appropriate Technology
The Task
The Task
Go two blocks for a newspaper
Solve 3x 21
Go a mile to get vegetables for dinner
Solve 3x 6 21 - 5x
Go to a play downtown
Solve .7x3 2.9x 17.3
Kutzler, Bernhard. CAS as Pedagogical Tools for
Teaching and Learning Mathematics. Computer
Algebra Systems in Secondary School Mathematics
Education, NCTM, 2003.
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The High School Student Perspective
Appropriate Technology
Appropriate Technology
The Task
The Task
Buy a sweatshirt at the Exeter Bookstore
Solve 3x 21
Buy shampoo at Walgreens
Solve 3x 6 21 - 5x
Take a tour of the New Hampshire coast
Solve .7x3 2.9x 17.3
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Calculator Allowed or Not?
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The Basics

• Pedagogical Use 1 What I Already Know is True

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The Idea of Function

• Manipulating Functions
• Variable vs. Parameter
• Variation y kxn
• Gravity Formula

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Parameters vs. Variables
Susan stands on top of a cliff in Portugal and
drops a rock into the ocean. It takes 3.4
seconds to hit the water. Then she throws
another rock up it takes 4.8 seconds to hit the
water. (a) How high is the cliff, to the
nearest meter? (b) What was the initial upward
velocity of her second rock, to the nearest
m/sec? (c) Which ocean did she drop the rock
into?
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Idea of Function

• Manipulating Functions
• Variable vs. Parameter
• Variation y kxn
• Gravity Formula
• Functions of several variables
• Combinations and Permutations
• Distance Formula

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Functions of Multiple Variables
For all positive integers x and y, if is
defined by x y (x y) 1, find (3 4) 5
If f (x, y) (x y) 1, find f ( f (3, 4), 5)
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Powers and Roots

• Pedagogical Use 2 There seem to be some more
  truths out there.
• Rationalize denominators.
• When should denominators be rationalized?
• Why should denominators be rationalized?
• Imaginary and complex numbers

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Rationalizing Denominators?
examples from UCSMP Advanced Algebra,
supplemental materials, Lesson Master 8.6B
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Powers and Roots

• Show that

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Is there something else out there?
What are the two things you have to look out for
when determining the domain of a function? What
does your calculator reply when you ask it the
following? a. 9 0 b.
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Polynomials and Rational Functions

• Change forms for equation
• What does factored form tell you?
• What does expanded form tell you?

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Polynomials

• The function f (x) -x3 5x2 kx 3 is
  graphed below, where k is some integer. Use the
  graph and your knowledge of polynomials to find
  k.

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Rational Functions The Old Rule

• Let f be the rational function
• where N(x) and D(x) have no common factors.
• If n lt m, the line y 0 (the x-axis) is a
  horizontal asymptote.
• If n m, the line is a horizontal
  asymptote.
• If n gt m, the graph of f has no horizontal
  asymptote.
• Oblique (slant) asymptotes are treated
  separately.
• n gt m 1 not dealt with at all.

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Rational Functions

• Expanded Form
• Factored Form
• Quotient-Remainder Form

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Rational Functions The New Rule

• Given a rational function f (x),
• Find the quotient and remainder.
• The quotient is the macro picture.
• The remainder is the micro picture -- it gives
  details near specific points.

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Rational Functions

• No need to artificially limit ourselves to
  expressions where the degree of the numerator is
  at most one more than the degree of the
  denominator.
• Analyze
• is just as easy as any other rational function.

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Rational Functions

• Analyze

Expanded form y-intercept is (0, 6)
vertical asymptote x -1
Factored form x-intercept at (1, 0)
Quotient-Remainder form Approaches f (x)
x2 - 4x
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Teaser Systems of Equations
Solve for x and y
Swokowski and Cole, Precalculus Functions and
Graphs. Question 11, page 538
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Other Extensions of the Curriculum

• Conic Sections
• Solutions to systems of conics
• Rotations of conics
• Exponential and Logarithmic Functions
• Logistic Functions
• Normal Functions

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Limitations

• Pedagogical Use 3 The Machine Doesnt Know
  Everything
• Youve gotta know the machine, and youve gotta
  know the mathematics.
• Real vs. Complex Numbers
• Let
• Graph y2 ( y1 (x) )

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Limitations

• Solve cant always solve algebraically.
• Trouble with radicals
• Variables both in and out of exponents

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Powers and Roots

• If ,
• what is the value of ?

Ohio Council of Teachers of Mathematics 2004
Contest, written by Duane Bollenbacher, Bluffton
College
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No exact solution
The teachers in the Valley Heights school
district receive a starting salary of 30,000 and
a 2000 raise for every year of experience. The
teachers in the Lower Hills district also receive
a starting salary of 30,000, but they receive a
5 raise for every year of experience. (a)
After how many years of experience will teachers
in the two school districts make the same salary
(to the nearest year)? (b) Is your answer in
(a) the only solution, or are there more? (c)
Ms. Jones and Mr. Jacobs graduate from college
and begin teaching at the same time, Ms. Jones in
the Valley Heights system and Mr. Jacobs in Lower
Hills. Will the total amount Mr. Jacobs earns in
his career ever surpass the amount Ms. Jones
earns? After how many years (to the nearest
year)?
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Variables in and out of exponents
From a question that arose while studying
compound interest A bank advertises a
certificate of deposit that pays 3.75 interest,
with an annual percentage yield (APY) of
3.80. How often is the interest compounded?
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Limitations

• Solve cant always solve algebraically.
• Trouble with radicals
• Variables both in and out of exponents
• Solve uses inverse functions.
• Inverse functions have limitations
• Non-linear functions as powers

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Limitations of Solve
Find all solutions to the equation Ohio
Council of Teachers of Mathematics 2002 Contest,
written by Duane Bollenbacher, Bluffton College
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Teaser Rational Numbers

• Is the number rational or irrational?

UCSMP Advanced Algebra, question 19, page 355
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A Deliberately Provocative Statement

• If algebra is useful only for finding roots of
  equations, slopes, tangents, intercepts, maxima,
  minima, or solutions to systems of equations in
  two variables, then it has been rendered totally
  obsolete by cheap, handheld graphing calculators
  -- dead -- not worth valuable school time that
  might instead be devoted to art, music,
  Shakespeare, or science.
• -- E. Paul Goldenberg
• Computer Algebra Systems in Secondary Mathematics
  Education

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Thank You!

• Michael Buescher
• Hathaway Brown School

For More CAS-Intensive work The USA CAS
conference http//www4.glenbrook.k12.il.us/USACAS/
2004.html