Transcript Computer Algebra Systems in Algebra II and Precalculus Courses

Integrating Computer Algebra
Systems into Algebra and
Precalculus Courses
Michael Buescher
Hathaway Brown School
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
What are Computer Algebra
Systems?
 Computer-based (Mathematica, Derive,
Maple) or Calculator-based (TI-89, TI-92,
HP-48, HP-49)
What are Computer Algebra
Systems?
 Computer-based (Mathematica, Derive, Maple) or
Calculator-based (TI-89, TI-92, HP-48, HP-49)
 Allow Symbolic Manipulation
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
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
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 can’t do something
is always an effective strategy [see Dress Code
discussion, session #1204]
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
What is your current attitude about
Computer Algebra Systems?
“This
“It gives
is madness.
lots of people
They won't
new life
learn
in
mathematics.
algebra. It will
It lets
cut off
them
careers
focus in
more
on the problem-solving
many fields."aspects rather
than the tedious computations."
-- Richard Askey,
University
of Wisconsin
at Madison
-- James
Schultz, Ohio
University
Madness
New Life
Quotes From:
Lisa Black, Robert Channick. “New Algebra: Batteries Required” Chicago Tribune, October
29, 2003 http://www.chicagotribune.com/news/local/chi-0310290205oct29,1,3428295.story
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
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
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 calculatorallowed
 Decide ahead of time what falls into what
categories!
Calculator Allowed or Not?

No Calculator
d  r  t , solve for r.
With CAS

g
m1  m 2
d2
, solve for d.

Solve t2 = 81


Solve x2 – 3x + 1 = 0
–4.9t2 + 23t + 19.2 = 12
Solve
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?
The Basics
 Pedagogical Use #1: What I Already
Know is True
The Idea of Function
 Manipulating Functions
 Variable vs. Parameter
– Variation: y = k·xn
1 2


h
t


gt  v0t  h0
– Gravity Formula:
2
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?
Idea of Function
 Manipulating Functions
 Variable vs. Parameter
– Variation: y = k·xn
1 2


h
t


gt  v0t  h0
– Gravity Formula:
2
 Functions of several variables
– Combinations and Permutations
– Distance Formula
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)
Teaser: Rational Numbers
Is the number
203
317
rational or irrational?
[UCSMP Advanced Algebra, question #19, page 355]
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
Rationalizing Denominators?
a)
1
3
4
b)
8 5
c)
d)
4
x
x
x 1
[examples from UCSMP Advanced Algebra, supplemental materials, Lesson Master 8.6B]
Powers and Roots
Show that
6 2

4
1 3
2
2
Powers and Roots
1
If x   3 ,
x
1
what is the value of x  2
x
2
?
[Ohio Council of Teachers of Mathematics 2004 Contest, written by Duane
Bollenbacher, Bluffton College]
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.
9
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
Polynomials and Rational
Functions
 Change forms for equation
 What does factored form tell you?
f x   x  22 x  13x  11x2  2 x  5
 What does expanded form tell you?
f x   6x5  43x4  119x3  187x2  91x  110
Polynomials
The function f (x) = -x3 + 5x2 + k∙x + 3 is
graphed below, where k is some integer. Use
the graph and your knowledge of polynomials
to find k.
Xscl = 1; Yscl = 1;
all intercepts are integers.
Rational Functions: The Old Rule
 Let f be the rational function
N x  an x n  an1 x n1    a1 x  a0
f x  

Dx  bm x m  bm1 x m1    b1 x  b0
where N(x) and D(x) have no common factors.
– If n < m, the line y = 0 (the x-axis) is a horizontal
asymptote.
– If n = m, the line y  an b is a horizontal asymptote.
m
– If n > m, the graph of f has no horizontal asymptote.
 Oblique (slant) asymptotes are treated separately.
Rational Functions
 Expanded Form:
2 x 2  13x  18
f x  
x3
 Factored Form:

x  2 2 x  9 
f x  
x  3
 Quotient-Remainder Form:
3
f x   2 x  7 
x3
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.
x  3x  4 x  6
f x  
x 1
3
 Analyze
2
is just as easy as any other rational function.
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.
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.
Teaser: Systems of Equations
Solve for x and y:
2 2  x  3  y

3 x  2y


2
 2 3
Swokowski and Cole, Precalculus: Functions and Graphs. Question #11, page 538
Other Extensions of the
Curriculum
 Conic Sections
– Solutions to systems of conics
– Rotations of conics
 Exponential and Logarithmic Functions
– Logistic Functions
– Normal Functions
Limitations
 Pedagogical Use #3: The Machine
Doesn’t Know Everything
 “You’ve gotta know the machine, and
you’ve gotta know the mathematics.”
 Real vs. Complex Numbers
– Let y1x   x and y2x   x2
– Graph y2 ( y1 (x) )
Limitations
 “Solve” can’t always solve algebraically.
– May get approximate answers
– Combination of Linear and Exponential
Functions
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)?
Limitations
 “Solve” can’t 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
Limitations of “Solve”
Find all solutions to the equation
2 x
2
x
 8 x  7
2
5 x  6

1
[Ohio Council of Teachers of Mathematics 2002 Contest, written by
Duane Bollenbacher, Bluffton College]
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.
Standardized Test Scores
Year
2000
2001
2002
2003
CTP
20.0
20.3
20.7
20.8
2001
2002
2003
2004
Mean score
Class
SAT-I
650-670 650-670 620-640 650-670
Median Score
Before CAS
With CAS
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!
Thank You!
Michael Buescher
Hathaway Brown School
For more in-depth work with CAS …
http://mathconf.exeter.edu/