Integrating Computer Algebra Systems into Algebra and Precalculus Courses

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Integrating Computer Algebra Systems into Algebra
and Precalculus Courses

• Michael Buescher
• Hathaway Brown School

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On The Same Page

• What are Computer Algebra Systems?
• What I have done
• Where you are
• The process of technological integration and
  curriculum transformation
• The need for by-hand calculations

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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 four years
• TI-89 for all, Mathematica for me
• Traditional curriculum, heavily influenced by
  College Board AP Calculus

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Reasons for decision to use CAS

• 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 Dress Code
  discussion, session 1204

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How much do you use Computer Algebra Systems

• (A) Not considering using CAS
• (B) Considering using it in some courses
• (C) Using it in some courses considering it
  for other courses
• (D) Using it in all courses

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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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Technology and Curriculum Change

• Foundation ?Integration? Transformation (Jenny
  Little)
• Foundation Focus on the tool -- learning
  software, teaching how to use the machine
• Integration Teacher develops activities to
  support status-quo curriculum
• Transformation Developing constructivism,
  collaboration, and communication learning shifts
  to areas not possible without the technology

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Foundation?Integration?Transformation

• After four years, still at the Foundation ?
  Integration transition, with occasional glimpses
  of Transformation
• Focus on pedagogy and assessment curriculum
  change is slower and depends more on external
  factors

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CAS vs. by-hand calculations

• There are some skills that are important for
  students to be able to do on their own
• Every test and most quizzes include two parts
  paper-and-pencil only and calculator-allowed
• Decide ahead of time what falls into what
  categories!

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Calculator Allowed or Not?
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Examples of CAS in Different Areas of the
Curriculum

• Look at a few topics from the perspective of
• Pedagogy
• Assessment sample questions
• From Integration to Curriculum Transformation?

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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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Teaser Rational Numbers

• Is the number rational or irrational?

UCSMP Advanced Algebra, question 19, page 355
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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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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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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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Teaser Palindromes

• 20022002 is a palindrome
• (reads the same backwards and forwards).
• Find exactly three natural numbers, each one of
  them a palindrome of at least two digits, whose
  product is 20022002.

By Duane Bollenbacher, Bluffton College From
his Puzzle Corner in Ohio CTM Newsletter, March
2003
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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.

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Rational Functions

• Expanded Form
• Factored Form
• Quotient-Remainder Form

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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 The New Rule

• Given a rational function f (x),
• Find the quotient and remainder.
• The macro picture looks like the quotient.
• The remainder gives you details near specific
  points.

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Rational Functions

• Find the equation of a rational function that
  meets the following conditions
• Vertical asymptote x 2
• Slant (oblique) asymptote y 3x 1
• y-intercept (0, 4)
• Show all of your work, of course, and graph your
  final answer. Label at least four points other
  than the
• y-intercept with integer or simple rational
  coordinates.

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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.
• May get approximate answers
• Combination of Linear and Exponential Functions

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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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Limitations

• Solve cant always solve algebraically.
• May get approximate answers
• Combination of Linear and Exponential Functions
• 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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Conclusions

• CAS use in Algebra II and Precalculus has been
  very successful, from both a teacher and a
  student perspective.
• Standardized Test Scores are not noticeably
  impacted.

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Standardized Test Scores
Before CAS With CAS
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Conclusions

• CAS use in Algebra II and Precalculus has been
  very successful, from both a teacher and a
  student perspective.
• Standardized Test Scores are not noticeably
  impacted.
• Never Go Back!

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Thank You!

• Michael Buescher
• Hathaway Brown School

For more in-depth work with CAS
http//mathconf.exeter.edu/