Transcript Interactive Algebra Using CAS Calculators and Software (TI

Interactive Algebra
Using TI-89 Calculators
and TI-InterActive! Software
NCTM Central Regional Conference
Minneapolis, Minnesota
November 12, 2004
Lynda Plymate, Ph.D.
[email protected]; math.smsu.edu/~lynda
and
David Ashley, Ph.D.
[email protected]; math.smsu.edu/faculty/ashley.html
Southwest Missouri State University
Department of Mathematics
Springfield, Missouri 65804-0094
Can Calculators and Computers Increase
Mathematical Reasoning?
"At home and at work, calculators and computers are "power tools" that
remove human impediments to mathematical performance – they
extend the power of the mind as well as substitute for it – by performing
countless calculations without error or effort.
Calculators and computers are responsible for a "rebirth of experimental
mathematics" (Mandelbrot 1994). They provide educators with
wonderful tools for generating and validating patterns that can help
children learn to reason mathematically and master basic skills.
Calculators and computers hold tremendous potential for mathematics.
Depending on how they are used, they can either enhance
mathematical reasoning or substitute for it, either develop mathematical
reasoning or limit it.”
Steen, L.A. (1999). Twenty questions about mathematical reasoning. In L.V. Stiff &
F.R. Curcio (Eds), Developing mathematical reasoning in grades K-12: 1999
Yearbook (pp. 270-285). Reston, VA : NCTM.
Technology Used
$49.95 Student Edition
$64.95 Teacher Edition
with Activity Book
$135 School Pricing
$150 Viewscreen Calculator
$375 Viewscreen Package
Computer Algebra System (CAS)
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Recursion
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Drugs and alcohol
Tim and Tom
Exact Arithmetic
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Fractions
Radicals
2D & 3D Grapher
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Linear Functions
Level Curves
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Parametric Graph
Numeric Solver
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Area of Trapezoids
Symbolic Algebra
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Solve equations and formulas
Function operations
Units and Conversions
Simulations
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Spaghetti triangle simulation
Matrices
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Polio Pioneers
Algebra Standard
Instructional programs from prekindergarten through
grade 12 should enable all students to—
• Understand patterns, relations, and functions
• Represent and analyze mathematical situations
and structures using algebraic symbols
• Use mathematical models to represent and
understand quantitative relationships
• Analyze change in various contexts
Cost of Balloons
Number
of
Balloons
Cost of
Balloons
(in cents)
1
20
2
40
3
60
4
80
5
?
6
?
7
?
Looking
for Patterns
Grade Level
Expectations in Missouri
Grade 3 (left)
Grade 6 (below)
Cost of Balloons
Number of Balloons
1
2
3
4
Cost of Balloons
(in cents)
20
40
60 80
5
n
?
?
Write an equation which
names the relationship
between the two variables
x and y for each of the
following two graphs.
Looking
for Patterns
Grade-8 Level
Expectation in Missouri
Different views and
explanations of the
rational function
2 x  11x  6
Looking
for Patterns
2
f ( x) 
x2
f ( x )  2 x  15 
36
x2
Grade-10 Level
Expectation in Missouri
Match the following scenarios with the graphs. Name
& label (with appropriate numbers) the selected axes.
1.
A 24 inch string when tied together can be made
to form a infinite number of rectangles, with the
area of the rectangle changing as the width of
the same rectangle is made to get larger and
larger.
a
2.
The temperature of the filling in a frozen cherry
pie increased dramatically when it is placed
inside a preheated oven, then tapered off to a
relatively steady hot temperature.
3.
The population of frogs decreased as the pond
became more polluted.
4.
The diameter of the cocoon increased rapidly at
first, then increased more slowly as the
caterpillar prepared to change into a butterfly.
5.
The temperature inside an oven increases when
it is turned on, and then fluctuates a bit as the
oven turns off and on briefly, trying to maintain a
preset temperature.
6.
The length of time it takes to paint the
gymnasium changed as the number of people
painting increased.
c
b
d
e
f
g
h
The box plots shown below represent the ratings given to the 257
episodes, in the seven seasons, of Star Trek: The Next Generation (top
plot is season 1 and bottom plot is season 7). These ratings, with 1 as
the best and 257 as the worst, were determined by Entertainment
Weekly magazine personnel. Use the center and spread in these plots
to defend your choice of the “best season” for this program.
Martha made the pattern shown below on her TI-89
calculator. One line in this pattern has equation
y = x - 1. Determine the equations of the other 7
lines in this pattern. Use the numerical limits on
the x-axis and y-axis as references for your lines.
6
-9
9
-6
Equal Pay for Equal Education?
Years of
Income Men
Income Women
Schooling
Median in $1,000
Median in $1000
6
17.0
12.2
8
21.2
13.8
12
26.8
18.6
13
31.4
22.0
14
32.3
24.8
16
40.4
29.3
18
47.3
35.0
Median Pay for Years of Schooling
Income (in $1000)
50.0
y = 2.431x + 0.7
R2 = 0.9565
40.0
Men
30.0
Women
20.0
Linear (Men)
Linear (Women)
10.0
y = 1.8908x - 1.2577
R2 = 0.9519
0.0
0
10
Years of Schooling
20
Name:_________________________
STUDENT-GENERATED EXPONENT RULES
Dr. Lynda Plymate
Confidence Level
1. __________________________________________
(1=No Confidence to 5=Very
Confident)
____________________________
2. __________________________________________
____________________________
3. __________________________________________
____________________________
4. __________________________________________
____________________________
5. __________________________________________
____________________________
6. __________________________________________
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7. __________________________________________
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8. __________________________________________
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Power Versus Exponential Functions
Lynda Plymate
a=2
L1•
:=seq(
.0
x,x,.1
1,6,.1
) .2
.3
.4
.5
.6
.7
.8
.9
1.0
1.1
1.2
f(x) := x a
g(x) := a x
f(x) = x 2
g(x) = 2 x
Power• Exponent•
:=f(L1)
.000 :=g(L1)
1.000
.010
.040
.090
.160
.250
.360
.490
.640
.810
1.000
1.210
1.440
1.072
1.149
1.231
1.320
1.414
1.516
1.625
1.741
1.866
2.000
2.144
2.297
Power vs. Exponential Functions
Year
1790
1800
1810
1820
1830
1840
1850
1860
1870
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
Census
3.9
5.3
7.2
9.6
12.9
17.1
23.2
31.4
39.8
50.2
62.9
76
92
105.7
122.8
131.7
151.3
179.3
203.3
226.5
248.7
2000
276.2
Year
1790
1800
1810
1820
1830
1840
1850
1860
1870
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
Census
3.9
5.3
7.2
9.6
12.9
17.1
23.2
31.4
39.8
50.2
62.9
76
92
105.7
122.8
131.7
151.3
179.3
203.3
226.5
248.7
276.2
Interactive Explorations
Using TI-89 Calculators and TI-InterActive! sliders
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Use recursion for interest
Story of Tim and Tom
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Explore parameters of functions
Function Parameters
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Explore area and perimeter of polygons
Area and Perimeter
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Use random numbers for simulation
Spaghetti Triangle
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Explore equations of circles
Circles
The Better Raise
Imagine that you have been offered the position of Assistant Manager for Kinkos. You
have been told that your beginning salary is to be $30,000 per year, but you must choose
between the following two salary increase plans:
Plan 1: $2000 raise per year
Plan 2: $500 raise every 6 months
Which plan is better? _____________________________________________
Are you sure about that? To verify your answer, prepare a spreadsheet similar to what is
shown below and complete 20 years accumulated pay. (OR use Better Raise slider)
Plan 1
Plan 2
$2000 raise per year
$500 raise per 6 months
Year
Period (6month)
Pay for
Period
Total
Accumulated
Pay
Pay for Period
Total Accumulated
Pay
1
0
$15,000.00
$15,000.00
$15,000.00
$15,000.00
1
1
$15,000.00
$30,000.00
$15,500.00
$30,500.00
2
2
$16,000.00
$16,000.00
2
3
$16,000.00
$16,500.00
3
4
3
5
PROBLEM: An offshore oil well is located in the ocean at point W, which is 8 miles
from the closest shore point A on a straight shoreline. The oil is to be piped from W
to a shore point B, which is 10 miles from A, by piping it on a straight line
underwater to some shore point P, between A and B, and then on to B through an
underground pipe along the straight shoreline. If the cost of laying the pipe is
$50,000 per mile under water and $20,000 per mile under land, where should point P
be located to minimize the cost of laying the pipe? (Note: Figure not drawn to scale)
ASSIGNMENT:
1. Explore this problem numerically on a spreadsheet by choosing different values for
the length of AP, PB, and WP, to compare and contrast the cost of laying the pipeline
for locations of P.
2. The exact solution to the problem can be found by assigning variables to quantities
that vary, finding an appropriate cost function and domain, and applying tools of
differential calculus. Complete the problem again in this manner.
1Bremigan,
E. G. (2004). Note: Figures not drawn to scale. Mathematics Teacher, 98(2), 74-78.
What Does Research Tell Us About Using
Technology for Mathematics Instruction?
In her 1997 meta-analysis of all U.S. research studies involving technology1, and her later 2001
meta-analysis of research involving CAS environments specifically2, M. Kathleen Heid reports
findings from multiple researchers about using technology for mathematics instruction. She
reflects on issues about the nature of technology use, learning issues, curriculum issues, and
teacher preparation issues. Some of her discussion and findings are listed below:
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The use of calculators does not lead to an atrophy of basic skills.
Symbolic manipulation skills may be learned more quickly in areas such as introductory
algebra and calculus after students have developed conceptual understanding through
the use of cognitive technologies.
The concepts-before-skills approach using CAS in algebra and calculus courses, and
the inductive-before-deductive investigatory approach in geometry have been tested.
Graphics-oriented technology may level the playing field for males and females.
CAS students were more flexible with problem solving approaches and more able to
perceive a problem structure.
CAS students were able to move between representations and make connections.
CAS students outperformed others on tests, with and without use of CAS during test.
5 of 7 researching experts (on a panel) reported better understanding by CAS students
1 Heid,
M. K. (1997). The technological revolution and the reform of school mathematics,
American Journal of Education, 106(1)5-61.
2 Heid, M. K. (2001). Research on mathematics learning in CAS environments.
Presented at the 11th annual ICTCM Conference, New Orleans.
Discussion Questions
Questions of Access:
• Use if all students don’t have the technology
• Use technology in and out of class
• Convince others of it’s value
Questions of Instruction:
• Focus on mathematics not how to use technology
• Concepts before skills (problem solving first)
• Balance CAS usage with mental and paper work
Questions of Assessment:
• New types of test questions and test format
• Match testing, instruction & learning strategies