Transcript NCTM_2006 - Michael Buescher`s Home Page

CAS In the Classroom
Test Questions that Challenge and Stimulate
Michael Buescher
Hathaway Brown School
[email protected]
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
How CAS Might Change a Test
Question
CAS
CAS
CAS
CAS
is irrelevant to the question
makes the question trivial
allows alternate solutions
is required for a solution
CAS Makes it Trivial
Simplify:
Combine like terms
12
Reduce a fraction
100
Simplify a radical
48
12a 5
or
100a 3
or
5
12
25 7
96 x y z
CAS Makes it Trivial
 Simplify:
 Combine like terms
 Reduce a fraction
 Simplify a radical
Expand:
Distribute
FOIL
Binomial Theorem
CAS Makes it Trivial
 Simplify:
 Combine like terms
 Reduce a fraction
 Simplify a radical
 Expand:
 Distribute
 FOIL
 Binomial Theorem
Factor:
Quadratic trinomials
Any polynomial!
Over the Rational,
Real, or Complex
Numbers
CAS Makes it Trivial
 Simplify:
 Combine like terms
 Reduce a fraction
 Simplify a radical
Solve Exactly:
 Expand:
 Distribute
 FOIL
 Binomial Theorem
 Factor:
 Quadratic trinomials
 Any polynomial!
 Over the Rational, Real,
or Complex Numbers
Linear Equations
Quadratic Equations
Systems of Equations
Polynomial, Radical,
Exponential, Logarithmic,
Trigonometric Equations
CAS Makes it Trivial
 Simplify:
 Solve Exactly:
 Combine like terms
 Reduce a fraction
 Simplify a radical
 Expand:
 Distribute
 FOIL
 Binomial Theorem
 Factor:
 Quadratic trinomials
 Any polynomial!
 Over the Rational, Real,
or Complex Numbers
*
 Linear Equations
 Quadratic Equations
 Systems of Equations
 Polynomial, Radical, Exponential,
Logarithmic, Trigonometric Equations
Solve Numerically:
Any equation you
might run across *
Maybe not ANY equation. More on that later.
CAS Makes it Trivial
 Simplify:
 Combine like terms
 Reduce a fraction
 Simplify a radical
 Expand:
 Distribute
 FOIL
 Binomial Theorem
 Factor:
 Quadratic trinomials
 Any polynomial!
 Over the Rational, Real,
or Complex Numbers
 Solve Exactly:
 Linear Equations
 Quadratic Equations
 Systems of Equations
 Polynomial, Radical, Exponential,
Logarithmic, Trigonometric Equations
 Solve Numerically:
 Any equation you might run across
Solve Formulas for
any variable
CAS Makes it Trivial
 Simplify:
 Combine like terms
 Reduce a fraction
 Simplify a radical
 Expand:
 Distribute
 FOIL
 Binomial Theorem
 Factor:
 Quadratic trinomials
 Any polynomial!
 Over the Rational, Real,
or Complex Numbers
 Solve Exactly:
 Linear Equations
 Quadratic Equations
 Systems of Equations
 Polynomial, Radical, Exponential,
Logarithmic, Trigonometric Equations
 Solve Numerically:
 Any equation you might run across
 Solve Formulas for any variable
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
CAS Is Irrelevant to the
Question
Graph a function
Fit a model to data
Questions involving only arithmetic, not
symbolic manipulation
Calculate a slope
Find terms of a sequence
Evaluate a function at a point
CAS Makes
the Question
Trivial
Dealing with Trivial Questions
Allow only Paper and Pencil for some tasks
See Bernhard Kutzler (2000). Two-Tier
Examinations as a Way to Let Technology In.
Modify the questions so that CAS becomes
irrelevant.
Use questions where CAS also gives
answers that are algebraically correct but
not applicable to the situation.
Paper and Pencil Questions
Important to have both specific and
general questions
Solve 4x – 3 = 8
AND
Solve y = m x + b for x
Solve x 2 + 2x = 15 AND
0
Solve 54 = 2(1 + r)3 AND
Solve a x 2 + b x + c =
Solve A = P e
rt
for r
Modifying Questions
Focus on the Process rather than
the Result
 The TI-89 says that
2i
1 7
  i
1  3i
10 10
(see right).
Show the work that proves it.
Focus on the Process
 Mr. Buescher is trying to save for Maple’s college education.
He has $23,000 put away now, and hopes to have
$100,000 in sixteen years. He correctly sets up the
equation
16
100,000  23,0001  r 
After setting up the equation, he totally blanks out on how
to solve it. Please help him put the steps in the right order.
A. Divide by 23,000; subtract 1; take 1/16 power.
B. Divide by 23,000; take 1/16 power; subtract 1.
C. Take 1/16 power; divide by 23,000; subtract 1.
D. Take 1/16 power; subtract 1; divide by 23,000.
E. He will never have $100,000 so don’t bother to solve.
Focus on the Process
 For which of the following equations
would it be appropriate to use
logarithms as part of your solution?
A.
B.
C.
D.
E.
5000 = 2000 (1 + r) 20
5000 = 2000 (1 + .098) t
5000 = 2000 x 2 – 2000x + 1000
5000 = x 2000
Logarithms are never appropriate.
Thinking Also Required
The force of gravity (F) between two
objects is given by the formula
m1  m2
F G
d2
where m1 and m2 are the masses of the two
objects, d is the distance between them, and G
is the universal gravitational constant.
Solve this formula for d
Thinking Also Required
Solve
logx28 = 4
Thinking Also Required
Thinking Also Required
CAS Allows
Alternate
Solutions
CAS Allows Alternate Solutions
Find x so that the matrix
5 x 
2 4 


does NOT have an inverse.
CAS Allows Alternate Solutions
Find x so that the matrix
5 x 
2 4 


does NOT have an inverse.
CAS Allows Alternate Solutions
Find x so that the matrix
5 x 
2 4 


does NOT have an inverse.
Polynomials
The function f (x) = -x 3 + 5x 2 + 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.
Thinking Required
The graph at right shows a
fourth-degree polynomial
with real coefficients, using
a somewhat unhelpful
viewing window. What (if
anything) can you conclude
about the other two roots
of the polynomial?
Xscl = 1; Yscl = 10
all intercepts are rational
When can you use “Solve”
If you can show with paper and pencil
that you can solve a simpler version, then
using “solve” when faced with real,
crunchy, ugly data is OK.
Sometimes, setting up the equation is the
more important piece.
What We Teach
The “Real World”
The Algebra
Problem
Situation
Algebraic
Model
Interpretation
Solution
Kutzler, Bernhard. “CAS as Pedagogical Tools for Teaching and Learning Mathematics.”
Computer Algebra Systems in Secondary School Mathematics Education, NCTM, 2003.
Linear Equations:
 The table and graph below show the voter turnout in
Ohio for Presidential Elections from 1980 to 2000 [source:
Ohio Secretary of State, http://www.sos.state.oh.us/sos/results/index.html] . The
regression line for this data is y = -.004582 x + 9.8311
where x is the year and y is the percentage of registered
voters who cast ballots (65% = .65)
Ohio Voter Turnout in Presidential Elections
Turnout
73.88%
73.66%
71.79%
77.14%
67.41%
63.73%
90%
85%
80%
75%
Turnout
Year
1980
1984
1988
1992
1996
2000
70%
65%
60%
55%
y = -0.004582x + 9.831148
50%
2
R = 0.494013
45%
40%
1976
1980
1984
1988
1992
Year
1996
2000
2004
[Continued]
Use the equation to predict the voter turnout
in 2004.
In what year (nearest presidential election) does
the line predict a voter turnout of only 50%?
 Multiple Choice. The slope of this line is about -.0046.
What does this mean?
(A)
(B)
(C)
(D)
The average voter turnout decreased by 0.46% per year.
The average voter turnout decreased by 0.46% every four years.
The average voter turnout decreased by .0046% per year.
There is very little correlation between the variables.
A Big Math Question
Presidential Press Conference, April 28, 2005. Graph from www.whitehouse.gov
A Big Math Question
If retirement benefits increase from
$14,800 to $17,750 over 50 years, what is
the average annual rate of increase?
If inflation is 3% per year, how much will
you need in 50 years to buy what $14,800
will buy today?
CAS Required
Questions that are
Inaccessible without CAS
The algebra is too complicated
The symbolic manipulation gets in the
way of comprehension
What is Factoring Anyway?
Factor x2
– 8x + 15
Factor x2
– 8x + 3
Factor x2
– 8x + 41
Factor 2x5 + 11x4 + 8x3 – 38x2 – 177x – 70
The Important Question
Convert from standard to factored form
What does factored form tell you about
the polynomial?
Which factored form tells you what you
want to know?
Which Factored Form?
Factor 2x5 + 11x4 + 8x3 – 38x2 – 177x – 70
Over Integers:
(2x – 5)(x 2 + 3x + 7) (x 2 + 5x + 2)
Over Reals (exact):
txt
2 x  52 x  17  52 x  17  5x 2  3x  7
4
Over Reals (approx.):
2(x – 2.5)(x + .438)(x + 4.562)(x 2 + 3x + 7)
Which Factored Form?
Factor 2x5 + 11x4 + 8x3 – 38x2 – 177x – 70
Over Complex with Rational Parts:
(2x – 5)(x 2 + 3x + 7) (x 2 + 5x + 2)
Over Complex with Real Parts (exact):
txt
2 x  52 x 




17  5 2 x  17  5 2 x   3  19 i 2 x  3  19 i
16

Over Complex with Real Parts (approx):
 2(x – 2.5)(x + .438)(x + 4.562)(x + 1.5 – 2.179i)(x + 1.5 + 2.179i)
CAS Required – Too Complicated
 Consider the polynomial
f x   2 x 5  7 x 4  5x 3  65x 2  23x  78
a. Sketch; label all intercepts.
b. How many total zeros does f (x) have? _______
c. How many of the zeros are real numbers? ______
Find them.
d. How many of the zeros are NOT real numbers?
______ Find them.
Rational Functions
Find a rational function that meets the
following conditions:
Two vertical asymptotes: at x = 3 and x = -¼
x-intercept (1, 0)
Approaches y = 2x + 3 as x  ∞
Symbolic Manipulation
gets in the way
Solve for x and y:
2 2  x  3  y 
2

 3x  2  y  2 3
Swokowski and Cole, Precalculus: Functions and Graphs. Question #11, page 538
Young Mathematicians
Discover Interesting Result
The Arithmetic Sequence
Question
Given an arithmetic sequence a with first
term 4 and common difference 1.2, …
Show that a5 + a8 = a10 + a3
Show that if m + n = j + k,
then
a m + an = a j + ak
The Arithmetic Sequence
Question
Given an arithmetic sequence a with first
term t and common difference d, …
Show that a5 + a8 = a10 + a3
Show that if m + n = j + k,
then
a m + an = a j + ak
CAS on Tests
Test some skills without CAS.
Modify questions to test process
understanding.
Encourage multiple solution routes, only
some including CAS.
Expand the expectations and the
curriculum.
Questions
Beyond CAS
Variables in the Base AND
in the Exponent
If a Certificate of Deposit pays 5.12% interest,
which corresponds to an annual rate of 5.25%,
how often is the interest compounded?
Variables in the Base AND
in the Exponent
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?
(b) Is your answer in (a) the only solution, or are there
more?
Even More Complicated
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.
(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)?
n
1  1.05n
30,000  30,000  2000n  1  30,000
2
1  1.05
Powers and Roots
Show that
6 2

4
1 3
2
2
What Mathematica Says
Powers and Roots
If
1
x 3 ,
x
what is the value of
1
x  2
x
2
?
[Ohio Council of Teachers of Mathematics 2004 Contest, written by Duane
Bollenbacher, Bluffton College]
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]
Thank You!
Michael Buescher
Hathaway Brown School
For More CAS-Intensive work:
The USA CAS conference
At the NCTM Regional Conference
Chicago, September 2006